Models of Genus One Curves
Mohammad Sadek
Fitzwilliam College, Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
December 2009

Declaration
This dissertation is the result of my own work and includes nothing which is the outcome
of work done in collaboration. No part of this dissertation has been submitted for any
other qualification.

Acknowledgements
I am indebted to my supervisor, Tom Fisher, for encouragement, guidance, inspiration,
and for suggesting the topic of the thesis to me.
I acknowledge with gratitude the financial support of Cambridge Overseas Trust. I
also wish to thank the Department of Pure Mathematics and Mathematical Statistics
of the University of Cambridge for funding my research trips.
Finally, I would like to thank my parents and sisters for all their love, support, and
patience.

Models of Genus One Curves
Mohammad Sadek
Summary
In this thesis we give insight into the minimisation problem of genus one curves defined
by equations other than Weierstrass equations. We are interested in genus one curves
given as double covers of P1, plane cubics, or complete intersections of two quadrics
in P3. By minimising such a curve we mean making the invariants associated to its
defining equations as small as possible using a suitable change of coordinates. We
study the non-uniqueness of minimisations of the genus one curves described above.
To achieve this goal we investigate models of genus one curves over Henselian discrete
valuation rings. We give geometric criteria which relate these models to the minimal
proper regular models of the Jacobian elliptic curves of the genus one curves above. We
perform explicit computations on the special fibers of minimal proper regular models of
elliptic curves. Then we use these computations to count the number of minimisations
of a genus one curve defined over a Henselian discrete valuation field. This number
depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point.
Finally, we consider the minimisation problem of a genus one curve defined over Q.

Contents
1 Introduction 1
2 Preliminaries 7
2.1 Genus one equations of degree n . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Models of curves and contraction . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Models of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Canonical sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Degree-n-models 18
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Singular Loci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Criteria for minimality 31
4.1 Canonical sheaves of degree-n-models . . . . . . . . . . . . . . . . . . . . 31
4.2 Constructing minimal degree-n-models . . . . . . . . . . . . . . . . . . . 36
4.3 Geometric criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Isomorphisms of degree-n-models 45
5.1 Isomorphic degree-n-models . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Degree-n- and degree-(n− 1)-models, n ≥ 3 . . . . . . . . . . . . . . . . 49
6 Computing in Emin 53
6.1 The components group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 The set ΦmK(E), m ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2.1 Reduction type I∗n, n ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . 55
6.2.2 Reduction types IV∗, III∗ and II∗ . . . . . . . . . . . . . . . . . . 60
vii
7 Counting minimal degree-n-models 65
7.1 Counting results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Constructing divisors and models . . . . . . . . . . . . . . . . . . . . . . 70
8 Counting models over Q 74
8.1 Fp-rational divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.2 Proof of Theorem 8.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.3 Counting global models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A Insoluble degree-n-models 92
A.1 Special fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
A.2 Insoluble degree-n-models, n = 2, 3 . . . . . . . . . . . . . . . . . . . . . 94
A.3 Insoluble degree-4-models . . . . . . . . . . . . . . . . . . . . . . . . . . 98
viii
Chapter 1
Introduction
Let E be an elliptic curve defined over a number field K. The problem of finding the
Mordell-Weil group E(K) has been a target for an enormous amount of research. It
is known that E(K) is a finitely generated abelian group. Furthermore, to determine
E(K) we only need to find E(K)/nE(K) for any integer n ≥ 2. The method of n-
descent is one of the methods which enable us to get a bound on E(K)/nE(K), n ≥ 2.
Indeed, the n-descent computes the n-Selmer group of E which contains E(K)/nE(K).
The difference between the two groups is the n-torsion of the Tate-Shafarevich group of
E/K.
An element of the n-Selmer group can be represented as a geometric object, namely
as a double cover of P1 ramified in four points when n = 2, a plane cubic curve when
n = 3, an intersection of two quadrics in P3 when n = 4. It follows from the definition
of the n-Selmer group that each of these curves has points everywhere locally. The
equations defining these genus one curves will be called genus one equations of degree
n. See §2.1 for the precise definitions of genus one equations and their invariants.
For using n-descent to search for points on E, we need the coefficients of the genus
one equations described above to be small. The solution has two parts: Reduction
and Minimisation. By reducing genus one equations, we mean reducing the size of the
coefficients by a unimodular linear change of coordinates, which does not change the
invariants. The problem of reduction is treated in [9] when n = 2, and in [11] when
n = 2, 3, 4. To minimise genus one equations, we need to make the associated invariants
smaller. The minimisation problem has been investigated intensively, for example see
[4] and [29] for n = 2, [14] for n = 3, and [30] for n = 4. An algorithmic approach to the
minimisation problem can be found in [11]. In this thesis we will be interested in the
minimisation question.
The question of minimisation has a local nature. More precisely, we minimise genus
1
one equations of degree n over local fields. Usually, there are many further details to
consider when the residue field is not algebraically closed. Therefore, we prefer studying
genus one equations over strict Henselisations of local fields because these fields have
algebraically closed residue fields. Thus we assume that our base field K is a Henselian
discrete valuation field with ring of integers OK and algebraically closed residue field
k. Moreover, since our genus one curves represent elements in n-Selmer groups, they
have rational points everywhere locally. Therefore, when we treat a genus one curve C
over K, we will assume that C(K) 6= ∅. It has been shown in [11] that if C is defined
by a genus one equation φ of degree n, and C(K) 6= ∅, then the minimal discriminant
associated to φ has the same valuation as the minimal discriminant associated to the
Jacobian elliptic curve.
Unlike elliptic curves, the minimisations of the genus one curves described above are
not unique under the action of the group Gn(OK) defined in §2.1. We give the following
example to illustrate that a genus one equation of degree 2 might have more than one
minimisation. We consider the elliptic curve E : y2 + y = x3 + x2 − 83x+ 244. We use
MAGMA, see [6], to perform 2-descent on E. We obtain the following genus one equation
φ : y2 = f(x) = 5x4 + 20x3 + 10x2 − 40x+ 25. We apply the G2(Z5)-transformation
(1, A), where A =
 1 3
0 2
 ,
to move the zeros of f(x)/5 to 0 and ∞. Then we are able to apply the G2(Q5)-
transformations given by (1/25, diag(5, 1)) and (1/25, diag(1, 5)), so that we have the
following three minimal genus one equations which lie in the same G2(Q5)-equivalence
class.
y2 = 27500x4 + 17000x3 + 4000x2 + 416x+ 16,
y2 = 1100x4 + 3400x3 + 4000x2 + 2080x+ 400,
y2 = 44x4 + 680x3 + 4000x2 + 10400x+ 10000.
It can be shown that the transformations stated above are the only ones relating these
genus one equations. Therefore, the genus one equations written above lie in three
distinct G2(Z5)-equivalence classes. In this thesis we study the non-uniqueness of min-
imisation by considering the problem geometrically. In fact, since E has reduction of
type IV∗ over Q5, Corollary 8.1.4 tells us that the number of G2(Z5)-equivalence classes
lying in the G2(Q5)-equivalence class containing φ is exactly 3.
The knowledge of all possible minimisations of a genus one curve can be exploited
to find rational points on elliptic curves. In fact, rational points on genus one curves,
2
defined by minimal genus one equations of degree n, can be expected to be reasonably
of smaller height, and therefore will be found more easily. Our results on the number of
minimisations will be used to study heights on such genus one curves in [26].
In Chapter 3 we generalise the terminology of minimal Weierstrass models for elliptic
curves to minimal degree-n-models for genus one curves. In fact, we define a degree-n-
model for a genus one curve C → Pn−1K to be a pair (C, α) consisting of a SpecOK-scheme
C defined by an integral genus one equation of degree n, and an isomorphism α from the
generic fiber CK of C onto C, where α is defined by an element in Gn(K). Now we show
why we insist on remembering the isomorphism α in our definition of a degree-n-model.
In the case C(K) 6= ∅, we pick P ∈ C(K). Then we identify the group structure on
(C,P ) with the group structure on the Jacobian elliptic curve E. The automorphism
group Aut(C) of C fits in an exact sequence
0→ E → Aut(C)→ Aut(E, 0)→ 0.
The first map is Q 7→ τQ, where τQ is the translation by Q. Let H be a hyperplane
section on C. We are interested in automorphisms λ of C such that λ∗H ∼ H. But
τ ∗QH ∼ H if and only if nQ = 0. Hence the elements of PGLn(K) that act fixed-
point-free on C correspond precisely to E[n](K). Therefore, we can have more than one
K-isomorphism between CK and C.
For example, we consider the elliptic curve E : y2 = x3 − x2 − 113x + 516 which
has a non-trivial 2-torsion point (−12, 0). Performing 2-descent on E will yield the
genus one equation φ : y2 = −3x4 − 56x3 − 399x2 − 1274x − 1519. We notice that
φ′ : y2 = −147x4 − 392x3 − 399x2 − 182x− 31 is a genus one equation which lies in the
G2(Q7)-equivalence class containing φ. The following two G2(Q7)-transformations carry
φ to φ′.
T1 = (1/7
2, diag(7, 1)), and, T2 = (1, A) where A =
 −7 1
−6 1
 .
Moreover, the transformation T1 is an element in G2(Q7) \ G2(Z7) while T2 is in G2(Z7).
The genus one equations φ and φ′ lie in the same G2(Z7)-equivalence class because they
are related via the transformation T2. Let C ′ be the SpecOK-scheme defined by φ′. We
distinguish between the two degree-2-models (C ′, α1) and (C ′, α2), where α1 and α2 are
defined by T1 and T2 respectively. The reason is that the special fibers of these two
models have different representatives in the special fiber of the minimal proper regular
model of the Jacobian.
It is known that an integral Weierstrass equation defining a Weierstrass modelW for
an elliptic curve E is minimal if the minimal desingularisation of W is isomorphic to the
3
minimal proper regular model of E. The analogous result for the case n = 2 has been
proved by Liu in [18]. Liu obtained his results using hyperelliptic involutions defined on
double covers of the projective line. We use our knowledge of invariant theory of genus
one curves to give geometric criteria for the minimality of genus one equations of degree
n for n ≤ 4, see Chapter 4.
In what follows we try to describe briefly how Liu obtained his results, and why we
did not use the hyperelliptic involution method to obtain our results. We suppose C
is a smooth curve defined by a minimal genus one equation. Let Cmin be the minimal
proper regular model of C. Since we assume C(K) 6= ∅, there exists a divisor of degree 2
on C. This divisor defines a separable morphism C → P1K of degree 2. The generator of
Gal(K(C)/K(P1K)) induces an automorphism σ of order 2 on C. We call σ a hyperelliptic
involution. Now σ extends to an automorphism σ˜ on Cmin. If Γ is an irreducible
component of multiplicity-1 in the special fiber of Cmin/〈σ˜〉, then we can find a model
P1 of P1K which is birational to Cmin/〈σ˜〉 in a neighborhood of the generic point of
Γ. Liu produced models for C by taking the normalisation of P1 in C. The reason
why we could not use this method to construct minimal degree-n-models for C, when
n = 3, 4, is that it only gives models for C → P1K , i.e., it gives models for C when C is
viewed as a double cover of the projective line. Moreover, even if we consider the models
produced as SpecOK-schemes, then the special fibers of these models either consist of
one irreducible component of multiplicity-m, m ≤ 2, or two irreducible components
of multiplicity-1. So, for example, we can not recover degree-n-models which contain
irreducible components of multiplicity-3 in their special fibers. But when n = 3, 4, there
are degree-n-models whose special fibers contain irreducible components of multiplicity-
3.
In Chapter 5 we give a necessary and sufficient condition for degree-n-models to
be isomorphic. In Chapter 6 we perform some explicit computations on the minimal
proper regular model of an elliptic curve. These computations are used to count minimal
degree-n-models for a soluble smooth genus one curve C → Pn−1K , up to isomorphism,
in Chapter 7. Liu proved that there is a bijection between minimal degree-2-models
for C → P1K , up to isomorphism, and the multiplicity-1 irreducible components in the
special fiber of the quotient of the minimal proper regular model by a hyperelliptic
involution, see ([18], Corollaire 5). Our results relate the number of minimal degree-n-
models for C → Pn−1K when n ∈ {2, 3, 4}, up to isomorphism, to the cardinality of a
finite set which depends only on the Kodaira Symbol of the Jacobian of C and on an
auxiliary rational point, see Theorem 7.1.1.
Finding all integral genus one equations of degree 2 which have the same invariants,
up to G2(Q)-equivalence, is an essential part of the 2-descent algorithm described by
4
Birch and Swinnerton-Dyer in [4]. The algorithm makes use of the fact that each G2(Q)-
equivalence class contains at least one reduced genus one equation of degree 2. Theorem
1 of [4] proves that reduced genus one equations of degree 2 are finite in number. The
problem of counting all the binary forms
f(x, z) = a0x
n + a1x
n−1z + . . .+ anzn,
with Z-coefficients and the same invariants, up to GL2(Z)-equivalence, is a classical
problem. For example, Theorem 1 of ([23], Chapter 18) asserts that the number of
such equivalence classes is finite. In a series of papers Bhargava studied the number
of many forms and tuples of forms, up to some equivalence relations defined over Z.
He presented interesting counting results for these equivalence classes. Moreover, he
defined composition laws on the equivalence classes of these objects. See [3] for some of
Bhargava’s results.
Our work contributes to the problem of counting forms and pairs of forms up to
relations defined over Z. In fact, we fix a Gn(Q)-equivalence class of genus one equations
of degree n, whose invariants are as small as possible, then we count the number of Gn(Z)-
equivalence classes within this Gn(Q)-equivalence class. We use arithmetic-geometric
methods to tackle this problem.
In Chapter 8 we get rid of the assumption that the residue field is algebraically
closed, and count the number of minimal degree-n-models for genus one curves defined
over p-adic fields. When the reduction of the Jacobian is split, we show that the number
of minimal degree-n-models for a genus one curve defined over a p-adic field is the same
as that number over the maximal unramified extension of the p-adic field. But maximal
unramified extensions are Henselian discrete valuation fields with algebraically closed
residue fields, and the number of models in this case is obtained in Chapter 7. Then
we work out how this number changes when the reduction of the Jacobian is non-split.
This step is a basic ingredient in counting minimal global degree-n-models for genus
one curves defined over Q. Our methods can be generalised easily to find the number
of minimal global degree-n-models for genus one curves defined over number fields with
class number one, as the problem reduces to considering genus one curves locally over
completions of number fields at non-archimedean places.
In Appendix A we work over complete discrete valuation fields with algebraically
closed residue fields. We treat the case when the genus one curve has no rational point
on it. We prove, using explicit computations, that the number of minimal degree-n-
models for such a curve is n when n ∈ {2, 3}. Then we give an example to show
that the number of minimal degree-4-models for insoluble genus one curves can become
arbitrarily large.
5
We have to mention that we proved our counting results only when the residue field
has characteristic greater than 3. We believe that similar results hold when the residue
characteristic is 3 but we have not checked the details. When the residue characteristic is
2, we believe that our counting results hold for minimal degree-n-models, when n = 3, 4,
but the problem needs deeper analysis. When the residue characteristic is 2 and n = 2,
we have to consider generalised genus one equations of degree 2, i.e., genus one equations
of the form φ : y2 + g(x)y = f(x), where deg g ≤ 2 and deg f ≤ 4. Moreover, the
invariants associated to φ are more complicated. When n = 2, Liu’s treatment of the
problem provided him with results when the residue characteristic is 2, see [18].
6
Chapter 2
Preliminaries
2.1 Genus one equations of degree n
In this section we will give a brief description of several genus one curves and the
equations defining them. We work over a perfect field K with algebraic closure K. We
write GK = Gal(K/K). We assume further that char(K) 6= 2, 3.
By a K-curve, or a curve over K, we mean a proper K-scheme that is geometrically
connected and of dimension 1.
Let C be a smooth curve of genus one over K. The function field of C will be denoted
by K(C). By a closed point P ∈ C we mean the GK-orbit of P considered as a geometric
point in C(K). The divisor group of C, called Div(C), is the free abelian group generated
by the closed points of C, i.e., a divisor D ∈ Div(C) is a formal sum
D =
∑
P∈C
nP (P )
with nP ∈ Z and nP = 0 for all but finitely many closed points P ∈ C. The divisor D
is said to be effective if nP ≥ 0 for every P ∈ C. The degree of D is defined by
degD =
∑
P∈C
nP [K(P ) : K].
Let CK = C ×K K. If we assume that the GK-orbit of P is {P1, . . . , PdP }, where dP =
[K(P ) : K], then we get a map
Div(C)→ Div(CK),
∑
nP (P ) 7→
∑
nP (
dP∑
i=1
Pi).
7
We define an action of GK on Div(CK) in the following way:
σ :
∑
P∈C(K)
nP (P ) 7→
∑
P∈C(K)
nP (P
σ).
Now we obtain Div(C) as the subgroup of GK-invariant divisors of Div(CK). In other
words, we have
Div(C) = Div(CK)
GK .
We will call a divisor D ∈ Div(C) a K-rational divisor on C.
The local ring of C at P will be denoted by OC,P . The ring OC,P is a discrete
valuation ring with maximal ideal mP and valuation νP .
Let f ∈ K(C)∗. Then we can associate to f the divisor
div(f) =
∑
P∈C
νP (f)(P ).
A divisor D ∈ Div(C) is said to be principal if D = div(f) for some f ∈ K(C)∗. Two
divisors D1, D2 are linearly equivalent, denoted D1 ∼ D2, if D1 −D2 is principal. The
divisor class [D] of a divisor D ∈ Div(C) is the set of all divisors linearly equivalent to
D. The quotient of Div(C) by the subgroup of principal divisors is the Picard group of
C, denoted Pic(C).
Let D ∈ Div(C) be of degree n > 0. Set
L(D) = {f ∈ K(C)∗| div(f) +D is effective} ∪ {0}.
Then Riemann-Roch Theorem implies that dimK L(D) = degD.
In what follows we aim to write explicit equations for the pair (C, [D]) when n =
1, 2, 3, 4, see [1]. We assume A is a Dedekind domain.
Genus one equations of degree 1
If degD = 1, then any effective rational divisor linearly equivalent to D is a rational
point P ∈ C(K). Let x, y ∈ K(C) be such that L(2(P )) and L(3(P )) have bases {1, x}
and {1, x, y} respectively. Note that x has a pole of exact order 2 at P, and y has a
pole of exact order 3 at P. There is a linear dependence relation between the 7 elements
1, x, y, x2, xy, x3, y2 in the 6-dimensional space L(6(P )). Furthermore, the coefficients
of x3 and y2 are non-zero. Rescaling x and y this linear dependence relation can be
assumed to be
y2 + a1xy + a3y = x
3 + a2x
2 + a4x+ a6, ai ∈ K. (2.1)
8
We will call the Weierstrass equation (2.1) a genus one equation of degree 1.
Two genus one equations of degree 1 with coefficients in A are A-equivalent, some-
times we will write G1(A)-equivalent, if they are related by the substitutions
x′ = u2x+ r, y′ = u3y + su2x+ t, r, s, t ∈ A, u ∈ A∗.
The group G1(A) is the group of all such transformations [u; r, s, t].We define det([u; r, s, t]) =
u−1.
We will write down the standard notations b2, b4, b6, b8, c4, c6, and the discriminant
∆ associated to equation (2.1), see ([27], Chapter III).
b2 = a
2
1 + 4a2,
b4 = 2a4 + a1a3,
b6 = a
2
3 + 4a6,
b8 = a
2
1a6 + 4a2a6 − a1a3a4 + a2a23 − a24, (2.2)
c4 = b
2
2 − 24b4
c6 = −b32 + 36b2b4 − 216b6,
∆ = −b22b8 − 8b34 − 27b26 + 9b2b4b6.
It is easy to verify that 1728∆ = c34 − c26.
Genus one equations of degree 2
If degD = 2, then we pick x, y ∈ K(C) such that L(D) and L(2D) have bases {1, x}
and {1, x, y, x2}. The 9 elements 1, x, x2, y, x3, xy, x4, x2y, y2 in the 8-dimensional space
L(4D) satisfy a linear dependence relation. Moreover, the coefficient of y2 is non-zero.
Therefore, (C, [D]) has equation
y2 + (α0x
2 + α1x+ α2)y = ax
4 + bx3 + cx2 + dx+ e. (2.3)
We will always assume that char(K) 6= 2. Therefore, by completing the square it
suffices to consider equations of the form
y2 = ax4 + bx3 + cx2 + dx+ e. (2.4)
Equation (2.4) is called a genus one equation of degree 2.
Two genus one equations of degree 2 with coefficients in A are A-equivalent, or
G2(A)-equivalent, if they are related by the substitutions
x′ = (m11x+m21)/(m12x+m22), y′ = µ−1y
9
where µ ∈ A∗, M = (mij) ∈ GL2(A). The group G2(A) is the group of all such trans-
formations [µ,M ]. We define det([µ,M ]) = µ det(M).
We associate the classical invariants I and J to equation (2.4), where
I = 12ae− 3bd+ c2,
J = 72ace− 27ad2 − 27b2e+ 9bcd− 2c3. (2.5)
We set c4 = 2
4I, c6 = 2
5J, and ∆ = (c34 − c26)/1728.
Genus one equations of degree 3
If degD = 3, then we pick x, y, z ∈ K(C) such that {x, y, z} is a basis for L(D).We write
a linear dependence relation between the 10 elements x3, y3, z3, x2y, x2z, y2x, y2z, z2x, z2y, xyz
in the 9-dimensional space L(3D). Therefore, (C, [D]) has a homogeneous ternary cubic
equation
ax3 + by3 + cz3 + a2x
2y + a3x
2z + b1y
2x+ b3y
2z + c1z
2x+ c2z
2y +mxyz = 0. (2.6)
Equation (2.6) is called a genus one equation of degree 3.
Two genus one equations of degree 3 with coefficients in A are A-equivalent, or
G3(A)-equivalent, if they are related by multiplying by µ ∈ A∗ and then substituting
x′ = m11x+m21y +m31z, y′ = m12x+m22y +m32z, z′ = m13x+m23y +m33z,
where M = (mij) ∈ GL3(A). The group G3(A) is the group of all such transformations
[µ,M ]. We define det([µ,M ]) = µ det(M).
We define the Hessian of a genus one equation φ(x, y, z) = 0 of degree 3 to be
H(φ) = −1/2× det

φxx φxy φxz
φyx φyy φyz
φzx φzy φzz
 .
Then we have
H(λφ+ µH(φ)) = 3(c4λ
2µ+ 2c6λµ
2 + c24µ
3)φ+ (λ3 − 3c4λµ2 − 2c6µ3)H(φ).
Moreover, we put ∆ = (c34 − c26)/1728, see [14].
10
Genus one equations of degree 4
If degD = 4, then we pick x1, x2, x3, x4 ∈ K(C) such that {x1, x2, x3, x4} is a basis for
L(D). Now we consider the 10 elements x21, x22, x23, x24, x1x2, x1x3, x1x4, x2x3, x2x4, x3x4 in
the 8-dimensional space L(2D). Therefore, (C, [D]) has equations
F1(x1, x2, x3, x4) = F2(x1, x2, x3, x4) = 0, where F1, F2 are quaternary quadratic forms.
(2.7)
We will call the equations (2.7) a genus one equation of degree 4.
Two genus one equations of degree 4 with coefficients in A are A-equivalent, or
G4(A)-equivalent, if they are related by
F ′1 = m11F1 +m12F2, F
′
2 = m21F1 +m22F2, M = (mij) ∈ GL2(A),
and then substituting
x′j =
4∑
i=1
nijxi, N = (nij) ∈ GL4(A).
The group G4(A) is the group of all such transformations [M,N ].We define det([M,N ]) =
det(M) det(N).
Let M1 and M2 be the 4 × 4 symmetric matrices of second partial derivatives of
F1 and F2 respectively. We associate a genus one equation φ of degree 2 to the genus
one equation {F1 = F2 = 0}, where φ : y2 = F (x, z) := det(M1x + M2z). Then we
set c4 = I, c6 = J/2 where I, J are the invariants associated to φ, see (2.5). We put
∆ = (c34 − c26)/1728.
Definition 2.1.1. Let Kn be the polynomial ring in the coefficients of a genus one
equation φ of degree n = 1, 2, 3, 4. A polynomial G ∈ Kn is an invariant of weight k if
G ◦ g = det(g)kG for all g ∈ Gn(K).
The following theorem indicates the properties of the invariants c4, c6, and ∆ defined
above.
Theorem 2.1.2. Let φ be a genus one equation of degree n = 1, 2, 3, 4. Let c4, c6, and
∆ be the associated invariants.
(i) The polynomials c4, c6,∆ ∈ Kn are invariants of weight 4, 6 and 12 respectively.
(ii) The equation φ defines a smooth curve Cφ of genus one if and only if ∆ 6= 0.
11
(iii) If char(K) 6= 2, 3, and ∆ 6= 0, then the Jacobian of Cφ has equation
y2 = x3 − 27c4x− 54c6.
Proof: See [15]. 2
Definition 2.1.3. Let R be a Dedekind ring with fraction field K. A genus one equation
φ of degree n with discriminant ∆ 6= 0 is
(i) integral if the defining polynomials have coefficients in R.
If R is a discrete valuation ring with normalised valuation ν, then φ is
(ii) minimal if it is integral and ν(∆) is minimal among all the valuations of the dis-
criminants of the integral genus one equations of degree n which are K-equivalent
to φ.
2.2 Models of curves and contraction
In this section we recall the definition of a model of a curve, introduce some well-known
models of smooth curves and record their basic properties. Then we define contraction
morphisms and give some results on their existence and uniqueness. References for this
are [5], [8] and [20].
2.2.1 Models of curves
Let R be a Dedekind domain with fraction field K. Put S = SpecR.
We recall that an S-scheme X is reduced at x ∈ X if the local ring OX,x has no
nilpotent elements. X is reduced if it is reduced at all its points. X is said to be integral
if it is reduced and irreducible. X is normal at x ∈ X if OX,x is integrally closed in
Frac(OX,x). We say X is normal if it is irreducible and normal at all of its points. The
scheme X is said to be regular at x ∈ X if OX,x is regular, i.e., dimOX,x = dimkmx/m2x,
where mx is the maximal ideal corresponding to x. X is regular if it is regular at all of
its points.
Definition 2.2.1. An S-curve is an integral, projective, flat, normal S-scheme f : X →
S of dimension 2.
We define what an S-model for a smooth curve over K is.
12
Definition 2.2.2. Let C be a smooth projective curve over K. An S-model for C is a
pair (C, i), where C → S is an S-curve and i : CK ∼= C is an isomorphism. A morphism
of S-models (C, i) → (C ′, i′) is an S-morphism α : C → C ′ such that i′ ◦ αK = i. An
S-model (C, i) for C dominates another model (C ′, i′) if there exists a morphism C → C ′
of S-models. We generally omit the explicit mention of i unless there is a danger of
confusion.
Now we introduce two of the most interesting models of smooth curves.
Definition 2.2.3. Let C be a smooth projective curve over K. A minimal proper regular
model for C is a regular S-model Cmin for C such that any domination map Cmin → C
to another regular S-model C for C is an isomorphism.
Theorem 2.2.4. Let C be a smooth projective curve over K. If C has positive genus,
then a minimal proper regular model Cmin for C exists and is unique up to unique
isomorphism. In particular, Cmin is dominated by all regular S-models for C.
Proof: See ([8], Theorem 3.9). 2
Definition 2.2.5. Let C be an S-model for a smooth curve C over K. A proper mor-
phism f : C ′ → C of S-models for C with C ′ regular is called a desingularisation of C.
We call a desingularisation morphism C˜ → C such that every other desingularisation
morphism C ′ → C factors uniquely through C ′ → C˜ → C a minimal desingularisation of
C. Moreover, C˜ is an S-model for C. By definition, if a minimal desingularisation exists,
then it is unique up to unique isomorphism.
Theorem 2.2.6. Let C be an S-model for a smooth curve C over K. If C has posi-
tive genus, then a minimal desingularisation C˜ → C exists and is unique up to unique
isomorphism.
Proof: See ([20], Proposition 9.3.36 (b)). 2
2.2.2 Contraction
Let K be a discrete valuation field with normalised valuation ν. We write OK for its
ring of integers. Fix a uniformiser t ∈ K and write k = OK/tOK for the residue field.
Put S = SpecOK . If X is an S-scheme, then we will denote its generic fiber by XK and
its special fiber by Xk.
13
Definition 2.2.7. Let C be an S-curve. Let (Γi)i∈I be the family of irreducible compo-
nents of the special fiber Ck. For a strict subset J ⊂ I, a contraction of the components
Γj, j ∈ J, in C consists of an S-morphism u : C → CJ of S-schemes such that
(a) For each j ∈ J , the image u(Γj) consists of a single point xj ∈ CJ , and
(b) u defines an isomorphism C −⋃j∈J Γj ∼−→ CJ −⋃j∈J xj.
Theorem 2.2.8. Assume that OK is Henselian. Let C be an S-curve. Let (Γi)i∈I be
the family of irreducible components of Ck. For a strict subset J ⊂ I, the contraction
u : C → CJ of the components Γj, j ∈ J, exists. Moreover, the morphism u is unique up
to unique isomorphism.
Proof: For the existence of u : C → CJ , see ([20], Theorem 8.3.36) or ([5], §6.7,
Proposition 4). For the uniqueness of u : C → CJ , see ([20], Proposition 8.3.28). 2
Let X be a scheme over S. We recall that a Cartier divisor D is a system {(Ui, fi)i},
where the Ui are covering open subsets of X, fi is the quotient of two regular elements
of OX(Ui), and fi|Ui∩Uj ∈ fj|Ui∩UjOX(Ui ∩ Uj)∗ for every i, j. Two systems {(Ui, fi)i}
and {(Vj, gj)j} represent the same Cartier divisor if on Ui ∩ Vj, fi and gj differ by a
multiplicative factor in OX(Ui ∩ Vj)∗.
To a Cartier divisorD we associate an invertible sheafOX(D) defined byOX(D)|Ui =
f−1i OX |Ui .We define the support of D to be the set of points x ∈ X such that OX(D)x 6=
OX,x, we denote it by SuppD. The set SuppD is a closed subset of X.
The Cartier divisor D is effective if it can be represented by {(Ui, fi)i} with fi ∈
OX(Ui). It is principal if it can be represented by a system {(X, f)}. An effective relative
Cartier divisor on X is an effective Cartier divisor on X which is flat over S when
considered as a closed subscheme of X. Linear equivalence is defined in the obvious
way. The group of isomorphism classes of Cartier divisors modulo linear equivalence is
denoted by CaCl(X).
A prime divisor on X is a closed integral subscheme of codimension one. A Weil
divisor is an element of the free abelian group generated by the prime divisors, i.e., we
write a Weil divisor D as
∑
niYi, where the Yi are prime divisors, the ni are integers, and
only finitely many ni are different from zero. If ni ≥ 0 for every i, then D is effective. A
Weil divisor is said to be principal if it can be written as
∑
νY (f).Y, where f ∈ K(X)∗,
and the sum is over all prime divisors of X and hence is finite. A Weil divisor D is locally
principal if X can be covered by open sets U such that D|U is principal for each U. We
define the linear equivalence as usual, and denote the group of isomorphism classes of
Weil divisors modulo linear equivalence by Cl(X).
14
Remark 2.2.9. If C is an S-curve, then CaCl(C) is isomorphic to the group Pic(C) of
isomorphism classes of invertible sheaves on C, see ([20], Corollary 7.1.19). Moreover,
the group of Cartier divisors on C is isomorphic to the group of locally principal Weil
divisors on C, see ([16], Chapter II, Proposition 6.11 and Remark 6.11.2).
If C is regular, then the group of Cartier divisors on C is isomorphic to the group of
Weil divisors on C, and CaCl(C) is isomorphic to Cl(C), see ([20], Proposition 7.2.16).
Proposition 2.2.10. Let C be an S-curve. Let L be an invertible sheaf on C generated
by global sections s0, . . . , sn. Let us consider the morphism f : C → PnS associated to
these sections. Let Γ be an irreducible component of Ck. Then f(Γ) is reduced to a point
if and only if L|Γ ' OΓ.
Proof: See ([20], Lemma 8.3.29). 2
The following theorem describes the contraction morphism explicitly. In fact, we use
it repeatedly in this thesis.
Theorem 2.2.11. Let C be an S-curve. Let (Γi)i∈I be the family of irreducible compo-
nents of Ck. Let D be a non-trivial effective relative Cartier divisor on C. Let J be the
set of all indices j ∈ I such that Supp(D) ∩ Γj = ∅. Then the canonical morphism
u : C → CJ := Proj(
∞⊕
m=0
H0(C,OC(mD)))
is a contraction of the components Γj, j ∈ J, and CJ is an S-curve.
Proof: Theorem 1 of ([5], §6.7) proves that the morphism u is a contraction of the
components Γj, j ∈ J, and that CJ is projective and normal. The proof that CJ is
integral is similar to the proof that CJ is normal with replacing the integral closeness
property with being an integral domain, see the proof of Lemma 2 of ([5], §6.7) and
([20], Proposition 2.4.17). The flatness is a direct consequence of the integrality of CJ
and that J 6= I, see ([20], Corollary 4.3.10). 2
Definition 2.2.12. Let C → S be a regular S-curve. Let Γ be an irreducible component
of Ck. Γ is called an exceptional divisor (or (−1)-curve) if there exist a regular S-curve
CΓ → S and a morphism u : C → CΓ of S-schemes such that u(Γ) is reduced to a point,
and that u : C−Γ ∼−→ CΓ−u(Γ) is an isomorphism, i.e., Γ can be contracted to a regular
point.
15
Example 2.2.13. Let C be a smooth curve defined over K. Assume that C has positive
genus. Let Cmin be the minimal proper regular model for C. Let C be an S-model for
C with minimal desingularisation C˜ → C.
Theorem 2.2.4 implies that there exists a unique morphism C˜ → Cmin as S-models
for C. Indeed, this morphism is the contraction morphism of the exceptional divisors in
C˜, see ([20], §10.1).
2.3 Canonical sheaves
In this section we define what a canonical sheaf is, and review some of its properties.
For a reference see ([20], Chapter 6).
Let f : X → Y be a morphism of schemes. Let ∆ : X → X ×Y X be the diagonal
morphism. The morphism ∆ gives an isomorphism ofX onto its image ∆(X). Moreover,
∆(X) is a closed subscheme of an open subset U of X ×Y X.
Definition 2.3.1. Let I be the sheaf of ideals of ∆(X) in U. We define the sheaf of
relative differentials of degree 1 of X over Y to be Ω1X/Y := ∆
∗(I/I2) on X. For any
r ≥ 1, we call the quasi-coherent sheaf ΩrX/Y := ∧rΩ1X/Y the sheaf of differentials of
order r.
Recall that an immersion of schemes is a morphism which is an open immersion
followed by a closed immersion.
Definition 2.3.2. Let f : X → Y be a morphism of schemes. Assume that there exists
an open subscheme V of Y such that f factors through a closed immersion i : X → V.
Let J be the sheaf of ideals defining i(X). The sheaf i∗(J /J 2) on X is called the
conormal sheaf of X in Y, and we denote it by CX/Y . This sheaf does not depend on the
choice of V.
Definition 2.3.3. Let F be a quasi-coherent sheaf on a scheme X. Assume moreover
that F is of constant rank ri on each connected component Xi of X. We define the
invertible sheaf detF by setting (detF)|Xi = ∧ri(F|Xi).
Example 2.3.4. Let X = SpecA[T1, . . . , Tn], where A is a Noetherian ring. Then
Ω1X/SpecA is locally free on X, and detΩ
1
X/SpecA = Ω
n
X/SpecA is free over OX , generated
by dT1 ∧ . . . ∧ dTn.
Definition 2.3.5. Let Y be a locally Noetherian scheme, and let f : X → Y be a
quasi-projective local complete intersection. Let i : X → Z be an immersion into a
16
scheme Z that is smooth over Y. We define the canonical sheaf of X → Y to be the
invertible sheaf
ωX/Y := det(CX/Y )∨ ⊗OX i∗(detΩ1Z/Y ).
This sheaf is independent of the choice of the decomposition X → Z → Y, up to
isomorphism.
Let f : X → Y be a smooth morphism of relative dimension d, i.e., for x ∈
X, dimxXf(x) = d. It is known that ωX/Y = ∧dΩ1X/Y .
The reason we are interested in canonical sheaves and not in sheaves of differentials
is that we are dealing with non-smooth schemes most of the time.
If A is a ring and {a1, . . . , an} is a sequence of elements of A. We say that it is a
regular sequence if a1 is not a zero divisor and for any i ≥ 2, ai is not a zero divisor
in A/(a1, . . . , ai−1). Now we state a lemma which enables us to compute the canonical
sheaves of S-curves.
Lemma 2.3.6. Let Y = SpecA be a Noetherian integral scheme, and let X be an
integral closed subscheme of Z = SpecA[T1, . . . , Tn] defined by an ideal generated by a
regular sequence F1, . . . , Fr with r ≤ n. Let us suppose that
∆ := det(
∂Fi
∂Tj
)1≤i,j≤r
is non-zero in K(X). Let ξ be the generic point of X.
(i) Let ti be the image of Ti in OX(X). Then
ωX/Y,ξ = (dtr+1 ∧ . . . ∧ dtn)OX,ξ.
(ii) As a subsheaf of ωX/Y,ξ, we have
ωX/Y = ∆
−1.(dtr+1 ∧ . . . ∧ dtn)OX .
Proof: See ([20], Corollary 6.4.14). 2
17
Chapter 3
Degree-n-models
In this chapter we define what we mean by a degree-n-model for a smooth genus one
curve C over a discrete valuation field K. Then we prove that if such a model is minimal,
then it is an S-model for C, see Definition 2.2.2. We conclude by describing the singular
loci of these models.
3.1 Definitions
Let R be a Dedekind domain with fraction field K. Put S = SpecR.
The S-scheme defined by an integral genus one equation φ of degree 1 is simply the
S-scheme C ⊂ P2S defined as follows
ProjR[x, y, z]/(φ : y2z + a1xyz + a3yz
2 − (x3 + a2x2z + a4xz2 + a6z3)), ai ∈ R.
Definition 3.1.1. Let φ be a genus one equation of degree 1. Let C be the elliptic
curve over K defined by φ. A degree-1-model for C is a pair (C, α) where C ⊂ P2S is an
S-scheme defined by an integral genus one equation of degree 1, and α : CK ∼= C is an
isomorphism defined by a K-equivalence of genus one equations of degree 1, i.e., α is
defined by an element of G1(K), see §2.1 for the definition of Gn(K).
An isomorphism β : (C1, α1) ∼= (C2, α2) of degree-1-models is an isomorphism β :
C1 ∼= C2 of S-schemes defined by an R-equivalence of genus one equations of degree 1,
i.e., β is defined by an element of G1(R), with βK = α−12 α1, see §2.1.
Note that our definition of a degree-1-model for an elliptic curve C is the definition
of a Weierstrass model for C as given in ([20], §9.4.4).
The S-scheme C defined by an integral genus one equation φ : y2 = F (x, z) of degree
2 is the scheme obtained by glueing {y2 = F (x, 1)} ⊂ A2S and {v2 = F (1, u)} ⊂ A2S via
18
x = 1/u and y = x2v. It comes with a natural morphism C → P1S given on these affine
pieces by (x, y) 7→ (x : 1) and (u, v) 7→ (1 : u).
The S-scheme defined by an integral genus one equation φ of degree n = 3, 4, is
simply the subscheme C ⊂ Pn−1S defined by φ.
Definition 3.1.2. (i) Let φ be a genus one equation of degree n ∈ {2, 3, 4}. Let
C → Pn−1K be the genus one curve defined by φ, where this morphism is a double
cover when n = 2, and it is an embedding when n ≥ 3.
A degree-n-model for C is a pair (C, α) where C → Pn−1S is an S-scheme defined
by an integral genus one equation of degree n, and α : CK ∼= C is an isomorphism
defined by a K-equivalence of genus one equations of degree n, i.e., α is defined
by an element of Gn(K).
(ii) An isomorphism of degree-n-models (C1, α1) ∼= (C2, α2) is an isomorphism β : C1 ∼=
C2 of S-schemes defined by an R-equivalence of genus one equations of degree n,
i.e., β is defined by an element of Gn(R), with βK = α−12 α1.
If there is no confusion, then we will omit mentioning the isomorphism α in the
degree-n-model (C, α) and write C instead.
Definition 3.1.3. Let R be a discrete valuation ring. A degree-n-model (C, α) for a
smooth genus one curve over K is said to be minimal if the defining genus one equation
of degree n of C is minimal, see Definition 2.1.3.
3.2 Normality
We will fix the following notations for the rest of this section unless otherwise stated.
K is a Henselian discrete valuation field with normalised valuation ν. We write OK for
its ring of integers. Fix a uniformiser t ∈ K and write k = OK/tOK for the residue
field. We will assume moreover that k is algebraically closed and that char(k) 6= 2. Set
S = SpecOK
In this section we find necessary and sufficient conditions for a degree-n-model for a
smooth genus one curve C to be normal, and hence to be an S-model for C.
If f(x1, . . . , xn) =
∑m
i=1 aix
l1i
1 . . . x
lni
n ∈ OK [x1, . . . , xn], then f˜(x1, . . . , xn) will denote
its image in k[x1, . . . , xn]. Moreover, ν(f) = min{ν(ai) : 1 ≤ i ≤ m}.
A Noetherian S-scheme X is said to be Cohen-Macaulay if OX,x is a Cohen-Macaulay
ring for every x ∈ X.
Let (A,m) be a regular Noetherian local ring of dimension d. Any system of gener-
ators of m with d elements is called a coordinate system for A.
19
Lemma 3.2.1. Let (A,m) be a regular Noetherian local ring.
(i) Suppose that f ∈ m\{0}. Then A/fA is regular if and only if f 6∈ m2.
(ii) Suppose that I is a proper ideal of A. Then A/I is regular if and only if I is
generated by r elements of a coordinate system for A, with r = dimA− dimA/I.
In other words, if and only if I is generated by r elements of m which are linearly
independent mod m2.
Proof: See ([20], Corollary 4.2.12 and Corollary 4.2.15). 2
We state the following lemma which we use throughout this section.
Lemma 3.2.2. Let C → S be a local complete intersection. Assume that CK is normal.
Then the following statements are true.
(i) C is normal if and only if C is regular at the generic points of Ck.
(ii) If Ck is reduced, then C is normal.
Proof: (i) Since C is a local complete intersection, it follows that it is a Cohen-
Macaulay scheme, see ([20], Corollary 8.2.18). Therefore, as a consequence of Serre’s
criterion for normality, it follows that C is normal if and only if it is normal at its points
of codimension 1, i.e., either closed points of the generic fiber, or the generic points of
the special fiber, see for example ([20], Corollary 8.2.24). Since CK is normal, we have
that C is normal if and only if C is normal at the generic points of its special fiber and
that is equivalent to regularity at these generic points.
(ii) See ([20], Lemma 4.1.18). 2
It is known that if C is a curve over K, then the normality of C coincides with the
regularity.
We need to define the concept of multiplicity of an irreducible component. Let X
be a locally Noetherian scheme over k. Let Γ be an irreducible component of X with
generic point ξ. Let mξ be the maximal ideal corresponding to ξ. Then the multiplicity
of Γ in X is the dimension d of OX,ξ/mξ as a k-vector space. Moreover, d = 1 if and
only if OX,ξ is reduced (or, equivalently, if X is reduced on a non-empty open subset
containing ξ), see ([20], Definition 7.5.6).
If C is an S-curve, then there is a normalised valuation νΓ of K(C) corresponding
to each irreducible component Γ of Ck. The multiplicity of Γ in Ck is equal to νΓ(t), see
([20], Exercise 8.3.3 (a)).
Let C be a genus one curve defined by a genus one equation φ of degree n = 1, 2,
and let C be the degree-n-model for C defined by φ.
20
If n = 1, then Ck is one of the following
(1) smooth cubic (2) nodal cubic (3) cuspidal cubic.
If n = 2, i.e., φ : y2 = f(x), then we can classify Ck according to the repeated roots
of f(x). Therefore, Ck is one of the following
(1) smooth quartic (2) nodal quartic (3) cuspidal quartic
(4) two intersecting lines (5) two tangent conics (6) double line.
The forms (1), (2), (3), (4), (5) and (6) above correspond to f(x) having no repeated
roots, one and only one double root, a cubic root, two double roots, a root of order 4,
and f(x) = 0 mod t respectively.
Now we state the conditions for C to be normal in the following two propositions.
Proposition 3.2.3. Let C be a smooth genus one curve over K defined by an integral
genus one equation φ of degree 1. Let C be the degree-1-model for C given by φ. Then
the model C is normal.
Proof: Since Ck consists of one irreducible component of multiplicity-1, in particular
Ck is reduced, and C is smooth, it follows that C is normal, see Lemma 3.2.2 (ii). 2
The following normality condition, when n = 2, can be found in ([18], Lemme 5).
Proposition 3.2.4. Let C be a smooth genus one curve over K defined by an integral
genus one equation φ : y2 = f(x) of degree 2. Let C be the degree-2-model for C → P1K
given by φ. The model C is normal if and only if ν(f) ≤ 1.
Proof:
(i) If t - f(x), then the defining equation y2 − f˜(x) = 0 of Ck has no square factor,
and so Ck is reduced, see ([20], Exercise 2.4.1). Since C is smooth, and hence it is
normal, and Ck is reduced, it follows that C is normal, see Lemma 3.2.2 (ii).
(ii) If t | f(x), then the maximal ideal corresponding to the generic point ξ of Ck
is mξ = 〈t, y〉. Lemma 3.2.2 (i) implies that C is normal if and only if Ck is
regular at ξ, and Lemma 3.2.1 (i) shows that the latter statement is equivalent to
y2 − f(x) 6∈ m2ξ . Since y2 ∈ m2ξ and t | f(x), whence C is normal if and only if
t || f(x), i.e., ν(f) = 1.
2
When C is a genus one curve defined by a genus one equation φ of degree n = 3, 4,
and C is the degree-n-model for C → Pn−1K defined by φ, the combinatorial possibilities
for the special fiber Ck increase.
21
We start with φ : F (x, y, z) = 0 of degree 3, then the special fiber Ck is one of the
following, see ([25], p. 266).
(1) smooth cubic (2) nodal cubic (3) cuspidal cubic
(4) conic + line (5) conic + tangent (6) line + double line
(7) three lines (8) three concurrent lines (9) triple line.
When φ : F (x1, x2, x3, x4) = G(x1, x2, x3, x4) = 0 is of degree 4, we will assume that
Ck is a curve, hence F˜ and G˜ are coprime. The special fiber Ck is in one of the forms
given below, see [12] or ([7], p. 46).
(1) smooth quartic (2) nodal quartic (3) cuspidal quartic
(4) two secant conics (5) two tangent conics (6) four lines (skew quadrilateral)
(7) four concurrent lines (8) cubic + secant line (9) cubic + tangent line
(10) conic + two lines not crossing on it (11) conic + two lines crossing on it
(12) conic + double line (13) double conic (14) double line + two lines
(15) triple line + line (16) two double lines (17) quadruple line.
Remark 3.2.5. Let C be a degree-4-model for a smooth genus one curve. We will need
explicit defining equations for Ck when Ck contains a component of multiplicity greater
than one. These equations can be obtained after applying transformations in G4(k). For
the whole list of defining equations for the special fiber Ck, see [12] or ([7], p. 46).
Ck Defining equations
conic + double line x1x3 = x1x4 + x
2
2 = 0
double conic x21 = x
2
2 + x
2
3 + x
2
4 = 0 which is k-equivalent to
x21 = x
2
2 + x3x4 = 0
double line + two lines x21 + x
2
2 = x1x3 + µx2x4 = 0, µ ∈ k
triple line + line x1x2 = x
2
1 + x2x4 = 0
two double lines x21 = x2x4 + µx1x3 = 0, µ ∈ k
quadruple line x21 = x
2
2 + µx1x3 = 0, µ ∈ k
22
Consider a genus one equation φ of degree 3 given by
φ : by3 + f1(x, z)y
2 + f2(x, z)y + f3(x, z) = 0, (3.1)
where f1(x, z) = b1x+ b3z, f2(x, z) = a2x
2+mxz+c2z
2, f3(x, z) = ax
3+a3x
2z+c1z
2x+
cz3.
Let C be the degree-3-model defined by φ. Assume that its special fiber Ck contains
an irreducible component of multiplicity-m, m ≥ 2. Using a matrix in GL3(OK) we can
assume that the defining equation of this multiplicity-m component is y = 0. This means
that min{ν(f2), ν(f3)} ≥ 1, ν(f1) = 0 when m = 2, and min{ν(f1), ν(f2), ν(f3)} ≥
1, ν(b) = 0 when m = 3.
Proposition 3.2.6. Let C be the smooth genus one curve over K defined by the integral
equation φ : F (x, y, z) = 0 given in (3.1). Let C be the degree-3-model for C → P2K given
by the same equation.
(i) If Ck contains only multiplicity-1 components, then C is normal.
(ii) If Ck contains a multiplicity-m component, m ≥ 2, whose defining equation is
y = 0, then C is normal if and only if ν(f3) = 1.
Proof: (i) Since C is normal and Ck is reduced, then C is normal, see Lemma 3.2.2
(ii).
(ii) Assume that Ck contains an irreducible component Γ : {y = 0} of multiplicity
m = 2. Using a matrix in GL3(OK) we can assume that Ck is defined by y2x = 0. The
maximal ideal corresponding to the generic point ξ of Γ is mξ = 〈t, y〉, and the maximal
ideal corresponding to the generic point ξ′ of {x = 0} is mξ′ = 〈t, x〉. Lemma 3.2.1 and
Lemma 3.2.2 imply that C is normal if and only if F (x, y, z) 6∈ m2ξ and F (x, y, z) 6∈ m2ξ′ .
Since ν(f2) ≥ 1, we have y3, y2, f2(x, z)y ∈ m2ξ . Therefore, F (x, y, z) 6∈ m2ξ if and only
if t || f3(x, z), i.e., ν(f3) = 1. Moreover, it is always true that F (x, y, z) 6∈ m2ξ′ because
ν(by3 + b3y
2z + c2yz
2 + cz3) = 0. Hence C is normal if and only if ν(f3) = 1.
Assume that Ck consists of a multiplicity-3 irreducible component Γ : {y = 0}. The
maximal ideal corresponding to the generic point ξ of Ck is mξ = 〈t, y〉. Since ν(f2) ≥ 1,
it follows that C is normal if and only if ν(f3) = 1. 2
Now we study the normality of degree-4-models for smooth genus one curves. Con-
sider a genus one equation φ of degree 4 given by F (x1, x2, x3, x4) = G(x1, x2, x3, x4) = 0,
where F and G are given by the following two integral equations respectively
a1x
2
1 + a2x1x2 + a3x1x3 + a4x1x4 + a5x
2
2 + a6x2x3 + a7x2x4 + a8x
2
3 + a9x3x4 + a10x
2
4,
b1x
2
1 + b2x1x2 + b3x1x3 + b4x1x4 + b5x
2
2 + b6x2x3 + b7x2x4 + b8x
2
3 + b9x3x4 + b10x
2
4,
(3.2)
23
where F˜ , G˜ are coprime, and do not define coplanar lines.
Proposition 3.2.7. Let C be the smooth genus one curve over K defined by the integral
equation φ given in (3.2). Let C be the degree-4-model for C → P3K given by the same
equation.
(i) If Ck contains a multiplicity-1 component Γ, then C is normal at Γ.
(ii) If Ck is a conic and a double line with F˜ = x1x3 and G˜ = x1x4 + x22, then C is
normal if and only if
ν(x4F (0, 0, x3, x4)− x3G(0, 0, x3, x4)) = 1.
(iii) Assume that Ck is a double conic with F˜ = x21 and G˜ = x22 + x3x4. Then C is
normal unless F (0, x2, x3, x4) ≡ µ(x22 + x3x4) mod t2, for some µ ∈ OK.
(iv) Assume that Ck contains a line Γ : {x1 = x2 = 0} of multiplicity-m, m ≥ 2, with
F˜ = q(x1, x2) and G˜ = x1x3 + µx2x4 + q
′(x1, x2), µ ∈ k. If ν(F (0, 0, x3, x4)) = 1,
then C is normal at Γ.
Proof: (i) If Ck contains a multiplicity-1 component Γ, then Ck is reduced at the generic
point ξ of Γ, but C is normal, hence C is normal at ξ, see ([20], Lemma 4.1.18).
Now we use Lemma 3.2.1 (ii) and Lemma 3.2.2 (i) to study the normality of C
at components of multiplicity greater than 1. The model C is normal if and only if
F,G 6∈ m2ξ , and F,G are linearly independent mod m2ξ , for every generic point ξ of Ck.
Note that the linear independence condition for F,G mod m2ξ , ξ ∈ C, is: for λ1, λ2 ∈
OK [x1, . . . , x4]ξ, if λ1F + λ2G ∈ m2ξ , then λ1, λ2 ∈ mξ.
(ii) Let ξ be the generic point of the double line {x1 = x2 = 0} in Ck, then mξ =
〈x1, x2, t〉. It is clear that F,G 6∈ m2ξ . Now we consider the linear independence of F
and G. If λ1F + λ2G ∈ m2ξ , then the fact that x1 and t are linearly independent mod
m2ξ implies that λ1x3 + λ2x4 ∈ mξ, i.e., λ1 ≡ µx4 mod mξ and λ2 ≡ −µx3 mod mξ for
some µ ∈ OK . Therefore, C is normal if and only if ν(f) = 1, where
f = x4(a8x
2
3 + a9x3x4 + a10x
2
4)− x3(b8x23 + b9x3x4 + b10x24).
(iii) Let mξ = 〈x1, x22 + x3x4, t〉 be the maximal ideal corresponding to the generic
point ξ of the conic. We have G 6∈ m2ξ . If a5x22 + a9x3x4 = tu(x22 + x3x4), u ∈ OK , then
F 6∈ m2ξ if and only if ν(a6x2x3 + a7x2x4 + a8x23 + a10x24) = 1, otherwise F 6∈ m2ξ if and
only if ν(a5x
2
2 + a6x2x3 + a7x2x4 + a8x
2
3 + a9x3x4 + a10x
2
4) = 1.
To investigate the linear independence of F and G, we note that if λ1F +λ2G ∈ m2ξ ,
then λ2 ∈ mξ. The reason is t and x22+x3x4 are linearly independent mod m2ξ . Therefore,
24
the condition we obtained from F 6∈ m2ξ implies that λ1 ∈ mξ, and hence we get linear
independence.
(iv) Assume that ξ is the generic point of Γ : {x1 = x2 = 0}. The ideal mξ is given
by 〈x1, x2, t〉. Since F˜ = q(x1, x2), and ν(a8x23 + a9x3x4 + a10x24) = 1, we have F 6∈ m2ξ .
Since G˜ = x1x3 + µx2x4 + q
′(x1, x2), where µ ∈ k, we have G 6∈ m2ξ because x1x3 6∈ m2ξ .
Therefore, we need to check the linear independence only. Let λ1, λ2 ∈ OK [x1, . . . , x4]ξ
be such that λ1F + λ2G ∈ m2ξ . Since x1, x2 and t are linearly independent mod m2ξ , it
follows that λ2 ∈ mξ. Moreover, as ν(a8x23 + a9x3x4 + a10x24) = 1, we get λ1 ∈ mξ, hence
F and G are linearly independent mod m2ξ . 2
The proof of the following lemma can be found in §2.5.1 of [30].
Lemma 3.2.8. Let C be the smooth genus one curve over K defined by the integral
equation φ given in (3.2). Let C be the degree-4-model for C → P3K given by the same
equation.
(i) Assume that F˜ and G˜ have a common factor. Then φ is not minimal.
(ii) Assume that Ck is a quadruple line with F˜ = x21 and G˜ = x22. Then either φ is not
minimal, or C(K) = ∅.
Proof: (i) Applying a transformation in GL4(OK) we can assume that x1 | F˜ , G˜. Now
we deduce that φ is not minimal by applying the transformation
1
t
F (tx1, x2, x3, x4) =
1
t
G(tx1, x2, x3, x4) = 0.
(ii) If ν(F (0, 0, x3, x4)) > 1, then φ is not minimal as we can apply the following trans-
formation
1
t2
F (tx1, tx2, x3, x4) =
1
t
G(tx1, tx2, x3, x4) = 0.
Similarly, if ν(G(0, 0, x3, x4)) > 1, then φ is not minimal.
Now we assume that ν(F (0, 0, x3, x4)) = ν(G(0, 0, x3, x4)) = 1. Consider the pair of
quadrics
F ′ =
1
t
F (tx1, tx2, x3, x4) and G
′ =
1
t
G(tx1, tx2, x3, x4).
If F˜ ′ and G˜′ have a common factor, then the genus one equation F ′ = G′ = 0 is
not minimal by (i). So we assume that F˜ ′ and G˜′ have no common factor. Let
(x1, x2, x3, x4) ∈ C(K). Clearing denominators, we can assume that min1≤i≤4 ν(xi) = 0.
Reducing F,G mod t, we have t | x1, x2. Reducing F,G mod t2, we get t | x3, x4, which
is a contradiction. Therefore, C(K) = ∅. 2
25
Theorem 3.2.9. Let φ be an integral genus one equation of degree n ∈ {1, 2, 3, 4}
defining a genus one smooth curve C over K. Let C be the degree-n-model defined by φ.
When n = 4, assume that C is not isomorphic to a degree-4-model whose special fiber is
of the form {x21 = x22 = 0}. If φ is minimal, then C is normal. In particular, C is an
S-model for C.
Proof: If Ck consists only of multiplicity-1 components, then Ck is reduced and hence
C is normal, see Lemma 3.2.2 (ii). Therefore, we only need to assume that φ is of degree
n, n ≥ 2, and Ck contains a component of multiplicity greater than 1.
For n = 2, let φ : y2 = f(x) be minimal with t | f(x). Then C is normal, since
otherwise ν(f) ≥ 2, see Proposition 3.2.4 (ii), and φ is not minimal as we can apply the
transformation y2 = 1
t2
f(x).
For n = 3, let φ : F (x, y, z) = 0 be a minimal genus one equation of degree 3 as
in equation (3.1). Let Ck contain a multiplicity-m component, m ≥ 2. It follows that
after using a matrix in GL3(OK), we can assume that ν(f2), ν(f3) ≥ 1. We claim that
ν(f3) = 1, and hence C is normal, see Proposition 3.2.6 (ii), since otherwise ν(f3) ≥ 2
and φ is not minimal because we can apply the transformation 1
t2
F (x, ty, z).
For n = 4, let φ be a minimal genus one equation of degree 4 given by F = G =
0, where F and G are given as in equation (3.2). Let Ck contain a multiplicity-m
component, m ≥ 2. We will go through the different cases of Proposition 3.2.7.
If Ck : {x1x3 = x1x4+x22 = 0}, then we claim that ν(x4F (0, 0, x3, x4)−x3G(0, 0, x3, x4)) =
1, and hence C is normal. To prove that claim, we assume on the contrary that the latter
valuation is greater than 1. We use a matrix in GL4(OK) to get rid of the x21, x1x2 and
x1x4-terms in F and of the x
2
1, x1x2 and x1x3-terms in G. We notice that in the equation
x4F (0, 0, x3, x4)− x3G(0, 0, x3, x4) = −b8x33 + (a8 − b9)x23x4 + (a9 − b10)x3x24 + a10x34,
we have min{ν(b8), ν(a8 − b9), ν(a9 − b10), ν(a10)} ≥ 2. We apply the transformation
x1 7→ x1 − a8x3 − a9x4, xi 7→ xi, i = 2, 3, 4, to get rid of the terms a8x23 and a9x3x4.
Thereafter, we obtain the genus one equation φ′ : F ′ = G′ = 0, where
F ′ = x1x3 + a5x22 + a6x2x3 + a7x2x4 + a10x
2
4,
G′ = x1x4 + x22 + b6x2x3 + b7x2x4 + b8x
2
3 + (b9 − a8)x3x4 + (b10 − a9)x24.
We deduce that φ′ is not minimal by applying the transformation
1
t2
F ′(t2x1, tx2, x3, x4) =
1
t2
G′(t2x1, tx2, x3, x4) = 0.
Assume that Ck : {x21 = x22 + x3x4 = 0}. If C is not normal, then ν(F (0, x2, x3, x4)−
µ(x22 + x3x4)) ≥ 2 for some µ ∈ OK , see Proposition 3.2.7 (iii). But then φ is not
26
minimal as we can apply the transformation
1
t2
(F (tx1, x2, x3, x4)− µG(tx1, x2, x3, x4)) = G(tx1, x2, x3, x4) = 0.
Now assume that Ck contains a line Γ : {x1 = x2 = 0} of multiplicity-m, m ≥ 2,
with F˜ = q(x1, x2) and G˜ = x1x3 + µx2x4 + q
′(x1, x2), where µ ∈ k. We claim that
ν(a8x
2
3 + a9x3x4 + a10x
2
4) = 1, and hence C is normal at Γ, see Proposition 3.2.7 (iv),
since otherwise φ is not minimal because we can apply the transformation
1
t2
F (tx1, tx2, x3, x4) =
1
t
G(tx1, tx2, x3, x4) = 0.
2
The following corollary is a direct consequence of Lemma 3.2.8 and Theorem 3.2.9.
Corollary 3.2.10. Let φ be an integral genus one equation of degree n ∈ {1, 2, 3, 4}
defining a genus one smooth curve C over K. Assume that C(K) 6= ∅. Let C be the
degree-n-model defined by φ. If φ is minimal, then C is normal.
3.3 Singular Loci
Let C be an S-curve with a smooth generic fiber. Then it is known that the normality of
C implies that there are only finitely many non-regular points on C, and all these points
are closed points in the special fiber, see for example ([8], p. 8). The set of non-regular
points of C will be called the singular locus of C, and we will denote it by Sing(C).
In this section we will compute the singular locus Sing(C) of a normal degree-n-model
C for a smooth genus one curve where n ∈ {1, 2, 3, 4}.
The set of zeros of a polynomial f ∈ OK [x1, . . . , xn] will be denoted V (f). If f ∈
OK [x1, . . . , xn], then we will write fi(x1, . . . , xn) = f(x1, . . . , xn)/ti.
Proposition 3.3.1. Let C be a smooth genus one curve over K defined by an integral
genus one equation φ of degree 1. Let C be the degree-1-model for C given by φ. Then
Sing(C) consists of one point at most.
Proof: This point is the node of Ck if C has multiplicative reduction, and it is the
cusp of Ck if C has additive reduction. 2
27
Proposition 3.3.2. Let C be a smooth genus one curve over K defined by an integral
genus one equation φ : y2 = f(x) of degree 2. Assume that φ defines a normal degree-2-
model C for C → P1K . Then
Sing(C) =
 {(x0, 0) : (x− x0)2|f˜(x)} if ν(f) = 0{(x0, 0) : x0 ∈ V (f˜1)} if ν(f) = 1
Proof: Assume that ν(f) = 0. Lemma 3.2.1 (i) implies that the singular locus of C
consists of points P = (x0, 0) ∈ C such that y2 − f(x) ∈ m2P = 〈x− x0, y, t〉2. Therefore,
if ν(f) = 0, then Sing(C) is the set {(x0, 0) : (x− x0)2 | f˜(x)}.
Now assume that ν(f) = 1. The maximal ideal corresponding to the generic point
ξ of Ck is mξ = 〈t, y〉. Therefore, Sing(C) consists of the points P = (x0, 0) such that
f˜1(x0) = 0. 2
Remark 3.3.3. Keep the notations of Proposition 3.3.2. If ν(f) = 0, then there are
at most two points in the singular locus of C. If ν(f) = 1, then there are at most four
points in the singular locus of C corresponding to the zeros of f˜1(x).
Proposition 3.3.4. Let C be a smooth genus one curve over K defined by an integral
genus one equation φ of degree 3 given by
F (x, y, z) := by3 + f(x, z)y2 + g(x, z)y + h(x, z) = 0.
Assume that φ defines a normal degree-3-model C for C → P2K .
(i) If Ck consists of l multiplicity-1 irreducible components, then Sing(C) consists of
one point at most when l = 1, and is contained in the set of intersection points of
these components when l ≥ 2.
(ii) If min{ν(g), ν(h)} ≥ 1, then Sing(C) = {(x0 : 0 : z0) : (x0, z0) ∈ V (h˜1(x, z))}.
Proof: (i) If Ck is a nodal cubic or a cuspidal cubic, then Sing(C) consists of the node
or the cusp respectively. If Ck consists of more than one multiplicity-1 component, then
each point of Ck is regular except possibly the intersection points of these components.
(ii) Now assume that min{ν(g), ν(h)} ≥ 1. The normality implies that ν(h) = 1,
see Proposition 3.2.6. The maximal ideal corresponding to the generic point of the
multiplicity-m component, m ≥ 2, is 〈y, t〉. We dehomogenise by setting z = 1. Let
P = (x0 : 0 : 1) ∈ C. The maximal ideal corresponding to P is mP = 〈x − x0, y, t〉.
Lemma 3.2.1 (i) implies that P ∈ C is non-regular if and only if F (x, y, 1) ∈ m2P .
Since y3, y2, g(x, 1)y ∈ m2P and ν(h) = 1, it follows that P is non-regular if and only
h˜1(x0, 1) = 0. 2
28
Remark 3.3.5. Keep the notations of Proposition 3.3.4. If Ck consists only of multiplicity-
1 components, then the number of points in the singular locus of C is three points at
most.
If Ck contains a multiplicity-m component, m ≥ 2, then the fact that h(x, z) is a
homogeneous polynomial of degree 3 implies that Ck has at most three points in its
singular locus corresponding to the factors of h˜1(x, z).
Proposition 3.3.6. Let C be a smooth genus one curve over K defined by an integral
genus one equation φ : F = G = 0 of degree 4 given as in equation (3.2). Assume that
φ defines a normal degree-4-model C for C → P3K .
(i) If Ck consists of l multiplicity-1 irreducible components, then Sing(C) consists of
one point at most when l = 1, and is contained in the set of intersection points of
these components when l ≥ 2.
(ii) If Ck is a conic and a double line with F˜ = x1x3 and G˜ = x1x4 + x22, then
Sing(C) = {(0 : 0 : x : y) : (x, y) ∈ V (h˜1(x3, x4))},
where h(x3, x4) = x4F (0, 0, x3, x4)− x3G(0, 0, x3, x4).
(iii) If Ck is a double conic with F˜ = x21 and G˜ = x22 + x3x4, then
Sing(C) = {(0 : xy : −x2 : y2) : (x, y) ∈ V (h˜1(x2, x4))},
where h(x2, x4) = F (0, x2x4,−x22, x24).
(iv) Assume that Ck contains a line Γ : {x1 = x2 = 0} of multiplicity-m, m ≥ 2, with
F˜ = q(x1, x2) and G˜ = x1x3 + µx2x4 + q
′(x1, x2) where µ ∈ k. Then
S ′ ⊆ Sing(C) ∩ Γ ⊆ S ′ ∪ {(0 : 0 : 0 : 1)},
where S ′ = {(0 : 0 : x : y) : (x, y) ∈ V (F˜1(0, 0, x3, x4))}.
Proof: (i) If Ck is a nodal or a cuspidal quartic, then Sing(C) consists of the node or
the cusp respectively. If Ck consists of more than one multiplicity-1 component, then
each point of Ck is regular except possibly the intersection points of these components.
Therefore, we assume that Ck contains a component of multiplicity greater than 1,
we want to find the points of Sing(C) which lie on this multiple component. Let P ∈ C.
Then Lemma 3.2.1 (ii) implies that P ∈ C is non-regular if and only if either F ∈ m2P ,
or G ∈ m2P , or F,G are linearly dependent mod m2P .
29
(ii) The maximal ideal corresponding to the generic point of the double line is
〈x1, x2, t〉. Let P be a closed point on the double line. Since x1 and t are linearly
independent, we have F ∈ m2P if and only if x3(P ) = 0 and ν(a10) ≥ 2, in other words
F ∈ m2P if and only if P = (0 : 0 : 0 : 1) and ν(a10) ≥ 2.
Similarly, G ∈ m2P if and only if P = (0 : 0 : 1 : 0) and ν(b8) ≥ 2.
Now for a point P = (0 : 0 : x : y) on the double line of Ck, we have F and G are lin-
early dependent if and only if (x, y) ∈ V (h˜1(x3, x4)), where h(x3, x4) = x4F (0, 0, x3, x4)−
x3G(0, 0, x3, x4), see Proposition 3.2.7 (ii). Note that if a point P makes either F or G
lie in m2P , then it is a point at which F and G are linearly dependent.
(iii) Let l(x2, x3, x4) = F (0, x2, x3, x4)−µ(a5x22+a9x3x4), where µ = 1 if a5 = a9, and
µ = 0 otherwise. We have F ∈ m2p if and only if P ∈ V (l˜1(x2, x3, x4))∩V (x22+x3x4), i.e.,
P = (0 : xz : −x2 : z2) where (x, z) ∈ V (h˜1(x2, x4)) and h(x2, x4) = F (0, x2x4,−x22, x24).
For any P ∈ C, we have G 6∈ m2P because x3x4(P ) 6∈ m2P for any P ∈ C. There are no
new non-regular points which can cause linear dependence because x22(P ) + x3x4(P ) 6∈
m2P for any P ∈ C.
(iv) Now let P be a point on the multiple line Γ. F ∈ m2P if and only if P = (0 : 0 :
x : y), where (x, y) ∈ V (F˜1(0, 0, x3, x4).
Consider G˜ = x1x3 + µx2x4 + q
′(x1, x2) where µ ∈ k. If µ 6= 0, then G 6∈ m2P for any
P ∈ Γ. If µ = 0, then the linear independence of t and x1 implies that G ∈ m2P if and
only if x3(P ) = 0 and G˜1(0, 0, x3(P ), x4(P )) ∈ mP . Therefore, G ∈ m2P if and only if
P = (0 : 0 : 0 : 1) and ν(b10) ≥ 2. We have no new non-regular points which can cause
linear dependence. 2
Remark 3.3.7. Assume that the special fiber Ck is given by x22 = x1x3+µx2x4 = 0, µ ∈
k. We have
S ′ ∪ S ′′ ⊆ Sing(C) ⊆ S ′ ∪ S ′′ ∪ {(0 : 0 : 0 : 1)},
where S ′ is as in Proposition 3.3.6 and S ′′ = {(x : 0 : 0 : y) : (x, y) ∈ V (F˜1(x1, 0, 0, x4)}.
Hence Sing(C) consists of five points at most.
In any other case, the above proposition implies that a degree-4-model for a smooth
genus one curve has at most four points in its singular locus. Indeed, if Ck consists
of multiplicity-1 components, then the number of points in Sing(C) is bounded by the
number of these components. If Ck is a conic and a double line as in (ii), then the
number of points in Sing(C) is at most three points corresponding to the factors of the
degree-3 polynomial h˜1 given in (ii). If Ck is a double conic as in (iii), then Sing(C)
consists of at most four points corresponding to the factors of the degree-4 polynomial
h˜1 of (iii). If Ck consists of a double line and two simple lines, then Sing(C) consists at
most of three points on the double line plus the intersection point of the simple lines.
30
Chapter 4
Criteria for minimality
In this chapter we will assume that K is a Henselian discrete valuation field with nor-
malised valuation ν. We write OK for the ring of integers. We fix a uniformiser t. The
residue field k = OK/tOK is not necessarily algebraically closed. Set S := SpecOK .
When we are dealing with degree-2-models for smooth genus one curves, we are going
to assume that char k 6= 2.
In this chapter we give geometric criteria for the minimality of normal degree-n-
models for smooth genus one curves. The main result introduced in this chapter is
stated in the following theorem.
Theorem 4.0.1. Let φ be an integral genus one equation of degree n = 1, 2, 3, 4. Assume
that φ defines a smooth genus one curve C overK, and that C(K) 6= ∅. Assume moreover
that φ defines a normal degree-n-model C for C. Let E/K be the Jacobian elliptic curve
of C, and Emin be the minimal proper regular model of E.
Then φ is minimal, see Definition 2.1.3, if and only if C˜ ∼= Emin, where C˜ → C is the
minimal desingularisation of C.
Theorem 4.0.1 is known for the case n = 1, see ([20], §9.4) or [8]. Moreover, Liu
gave a proof for the case n = 2, see ([18], Proposition 8 (b)). We will give a proof which
works for n = 1, 2, 3, 4.
4.1 Canonical sheaves of degree-n-models
Let φ be an integral genus one equation of degree n = 1, 2, 3, 4. Assume that φ defines
a smooth genus one curve C/K whose Jacobian elliptic curve is E. Assume moreover
that C(K) 6= ∅. Then C ∼=K E, whence the minimal proper regular model Cmin of C
31
is isomorphic to the minimal proper regular model Emin of E. For this reason we will
dispense with Cmin and write Emin from here on.
The following proposition describes the canonical sheaf ωEmin/S of E
min.
Proposition 4.1.1. Let E/K be an elliptic curve. Let Emin be its minimal proper
regular model. Then the canonical sheaf ωEmin/S of E
min is a trivial line bundle on Emin.
In other words, there exists ω0 ∈ H0(Emin, ωEmin/S) such that ωEmin/S = ω0OEmin .
Proof: See ([8], Example 7.7). 2
If C is an S-model for a smooth genus one curve C, then the canonical sheaf ωC/S
of C satisfies ωC/S|C = ωC/K , see ([20], Theorem 6.4.9 (b)). Moreover, the restriction of
the canonical sheaf ωC/S on C gives a canonical injection H0(C, ωC/S) ↪→ H0(C, ωC/K),
see ([20], Corollary 9.2.25 (a)).
Lemma 4.1.2. Let C be a normal degree-n-model for a smooth genus one curve C/K
with minimal desingularisation g : C˜ → C. Assume that C(K) 6= ∅. Let Emin be the
minimal proper regular model of the Jacobian E of C. Then
H0(Emin, ωEmin/S) = H
0(C˜, ωC˜/S) ⊆ H0(C, ωC/S).
Proof: Since C˜ and Emin are two regular S-curves with a contraction map C˜ →
Emin, we have H0(Emin, ωEmin/S) = H
0(C˜, ωC˜/S) as subgroups of H0(E,ωE/K), see ([20],
Corollary 9.2.25 (b)).
Let F be the divisor such that C˜ \F ∼= C \g(F ). Then we have the following relations
in H0(E, ωE/K) :
H0(C˜, ωC˜/S) ⊆ H0(C˜ \ F, ωC˜/S) = H0(C \ g(F ), ωC/S) = H0(C, ωC/S),
the second equality holds because g(F ) has codimension 2 in C, see ([20], Lemma 9.2.17
(a)). 2
In the following proposition we compute the canonical sheaf of a degree-n-model for
a smooth genus one curve.
Proposition 4.1.3. Let φ be an integral genus one equation of degree n = 1, 2, 3, 4.
Assume that φ defines a smooth genus one curve C/K. Assume moreover that φ defines
a normal degree-n-model C for C. Then ωC/S = ωOC, where ω ∈ H0(C, ωC/K) is
(i) if n = 1 and φ : y2 + a1xyz + a3yz
2 = x3 + a2x
2z + a4xz
2 + a6z
3, then
ω =
du
2v + (a1u+ a3)
, where u = x/z, v = y/z ∈ K(C),
32
(ii) if n = 2 and φ : y2 = f(x, 1), then
ω =
dx
2y
,
(iii) if n = 3 and φ : F (x, y, z) = 0, then
ω =
du
∂F/∂v
, where u = x/z, v = y/z ∈ K(C),
(iv) if n = 4 and φ : F1(x1, x2, x3, x4) = F2(x1, x2, x3, x4) = 0, then
ω =
du
∂F1
∂w
∂F2
∂v
− ∂F1
∂v
∂F2
∂w
, where u = x2/x1, v = x3/x1, w = x4/x1 ∈ K(C).
Proof: According to the definition of the canonical sheaf, see Definition 2.3.5, we have
to check first that C is a local complete intersection. In fact, C is a global complete
intersection over S. That follows directly from the fact that C can be embedded in P2S
for n = 1, 2, 3, and it can be embedded in P3S as an intersection of a pair of quadrics for
n = 4.
For n = 1, 3, let V1 be the open affine subset of C obtained by setting z = 1. Then
we can write OC(V1) as
OC(V1) = OK [u, v]/(bv3 + f1(u, 1)v2 + f2(u, 1)v + f3(u, 1)),
where b = 0 and f1(u, 1) = 1 when n = 1. Therefore, ωV1/S = ωOV1 , see Lemma 2.3.6.
Let V2 be the open affine subset of C given by setting y = 1. Set r = x/y, s = z/y. Then
OC(V2) = OK [r, s]/(g(r, s)) where
g(r, s) = b+ f1(r, s) + f2(r, s) + f3(r, s).
Hence ωV2/S = ω
′OV2 , where ω′ = ds∂g/∂r .
We note that r = u/v, s = 1/v in K(C), and ds = − 1
v2
dv in H0(C, ωC/K). Therefore,
ω′ = ω.
For n = 2, the first affine piece V1 satisfies OC(V1) = OK [x, y]/(y2 − f(x, 1)) and
therefore ωV1/S = ωOV1 . Let V2 be the open affine subset of C given by s2 = f(1, r),
where r = 1/x, y = x2s. Then OC(V2) = OK [r, s]/(s2 − f(1, r)) and ωV2/S = ω′OV2 ,
where ω′ = dr
2s
. Since r = 1/x ∈ K(C) and dr = − 1
x2
dx ∈ H0(C, ωC/K), it follows that
ω′ = ω.
Now as C = V1 ∪ V2, we have ωC/S = ωOC.
33
The proof is similar for n = 4. Let V1 be the open subset of C given by x1 = 1. We
have
OC(V1) = OK [u, v, w]/(F1(1, u, v, w), F2(1, u, v, w)),
hence ωV1/S = ωOV by Lemma 2.3.6.
Now let V2 be the open subset of C given by x2 = 1. Set q = x1/x2, r = x3/x2, s =
x4/x2. Then OC(V2) = OK [q, r, s]/(F1(q, 1, r, s), F2(q, 1, r, s)).
Therefore, ωV2/S is generated by the rational differential
ω′ :=
dq
∂F1
∂s
∂F2
∂r
− ∂F1
∂r
∂F2
∂s
.
Using the relations
q = 1/u, r = v/u, s = w/u, dq = − 1
u2
du in K(C) and H0(C, ωC/K),
we find that ω′ = ω. As C = V1 ∪ V2, we have ωC/S = ωOC. 2
Recall the definitions of the groups Gn(K) from §2.1. Now we state the following
corollary of Proposition 5.19 in [15].
Corollary 4.1.4. Let φ1, φ2 be two K-equivalent integral genus one equations of degree
n = 1, 2, 3, 4, where φ1 = g.φ2, g ∈ Gn(K). Assume that φ1, φ2 define smooth genus
one curves whose Jacobian elliptic curve is E/K. Assume moreover that φ1, φ2 define
two normal degree-n-models C1, C2. If ωCi/S = ωiOCi , i = 1, 2, where ωi is defined as in
Proposition 4.1.3, then ω2 = α(det g)ω1 as elements in H
0(E, ωE/K), where α ∈ O∗K .
Proof: The genus one equation φi defines a smooth genus one curve Ci, i = 1, 2. The
element g ∈ Gn(K) defines an isomorphism γ : C1 ∼= C2. The isomorphism γ satisfies
γ∗ω2 = (det g)ω1, see ([15], Proposition 5.19). 2
Corollary 4.1.5. Let φ1, φ2 be two K-equivalent integral genus one equations of degree
n = 1, 2, 3, 4, with corresponding discriminants ∆1,∆2. Assume that φ1, φ2 define smooth
genus one curves whose Jacobian elliptic curve is E. Assume moreover that φ1, φ2 define
two normal degree-n-models C1, C2. If ωCi/S = ωiOCi , i = 1, 2, then
∆1ω
⊗12
1 = λ∆2ω
⊗12
2 ∈ H0(E, ωE/K)⊗12, where λ ∈ O∗K .
Proof: Assume that φ1 = g.φ2 where g ∈ Gn(K). Since ∆1 = (det g)12.∆2, see Theorem
2.1.2 (i), and ω1 = α(det g)
−1ω2, where α ∈ O∗K , see Corollary 4.1.4, we have ∆1ω⊗121 =
α12∆2ω
⊗12
2 . 2
34
Let φ1 be a minimal genus one equation of degree n = 1, 2, 3, 4. Let φ2 be a genus
one equation K-equivalent to φ1. Let ω1, ω2 be as in Corollary 4.1.4. We will call the
integer m such that ω2 = ut
−mω1, u ∈ O∗K , the level of φ2, and denote it by level(φ2).
Corollary 4.1.5 implies that the level of an integral genus one equation of degree
n does not depend on the choice of the minimal genus one equation φ1. It follows
immediately that an integral genus one equation φ of degree n is minimal if and only if
level(φ) = 0.
Note that according to Corollary 4.1.5, we have ν(∆2) = ν(∆1)+12 level(φ2). Hence
∆2 = u
−1t12 level(φ2)∆1.
Lemma 4.1.6. Let C be an S-model for a smooth genus one curve C. Then we have
H0(C,OC) = OK . In particular, if ωC/S = ωOC, then H0(C, ωC/S) = ωOK .
Proof: The OK-module H0(C,OC) is integral over OK , see ([20], Proposition 3.3.18).
Moreover, H0(C,OC) is contained in OC(C), but OC(C) = K as C is geometrically
integral, see ([20], Corollary 3.3.21). 2
Lemma 4.1.7. Assume that φ1, φ2 are two genus one equations of degree n defining two
normal degree-n-models C1, C2 for a smooth genus one curve C. Let E be the Jacobian
elliptic curve of C. Then we have H0(C1, ωC1/S) ⊆ H0(C2, ωC2/S) as sub-OK-modules of
H0(E, ωE/K) if and only if ν(∆(φ1)) ≤ ν(∆(φ2)). Moreover, the equality of the two
submodules holds if and only if φ1 and φ2 have the same level.
Proof: Let ωCi/S = ωiOCi , ωi ∈ H0(E, ωE/K), i = 1, 2. The assumptionH0(C1, ωC1/S) ⊆
H0(C2, ωC2/S) is equivalent to ω1OK ⊆ ω2OK , by Lemma 4.1.6, i.e., ω1 ∈ ω2OK .
Since ∆(φ1)ω
⊗12
1 = λ∆(φ2)ω
⊗12
2 for some λ ∈ O∗K , see Corollary 4.1.5, it follows that
ω1 ∈ ω2OK is equivalent to ∆(φ2) ∈ ∆(φ1)OK , i.e., ν(∆(φ1)) ≤ ν(∆(φ2)).
The equality of the sub-OK-modules H0(C1, ωC1/S) = H0(C2, ωC2/S) means that
ω1OK = ω2OK as OK-modules, i.e., ω1 ∈ ω2O∗K . The latter statement means that
φ1 and φ2 have the same level. 2
The following lemma compares the generators of the canonical sheaves of Emin and
C when Emin ∼= C˜.
Lemma 4.1.8. If f : Emin → C is a contraction morphism, where C is a degree-
n-model for a smooth genus one curve, then f∗ωEmin/S = ωC/S. In other words, if
ωEmin/S = ω0OEmin and ωC/S = ωOC, then ω0 ∈ ωO∗K .
Proof: See ([20], Corollary 9.4.18 (b)). 2
Now we will prove the first part of Theorem 4.0.1.
35
Proposition 4.1.9. Let φ be an integral genus one equation of degree n = 1, 2, 3, 4.
Assume that φ defines a smooth genus one curve C over K, and that C(K) 6= ∅. Assume
moreover that φ defines a normal degree-n-model C for C. Let E/K be the Jacobian
elliptic curve of C, and Emin be the minimal proper regular model of E.
If C˜ ∼= Emin, where C˜ → C is the minimal desingularisation of C, then φ is minimal.
Proof: The assumption C˜ ∼= Emin means that C is obtained from Emin by contracting
the components different from those of Ck, thus we have a contraction morphism f :
Emin → C.
We are going to prove that if C ′ is another normal degree-n-model for C given by a
genus one equation φ′ of degree n, then
H0(C, ωC/S) ⊆ H0(C ′, ωC′/S), (4.1)
hence ν(∆(φ)) ≤ ν(∆(φ′)), see Lemma 4.1.7, therefore φ is minimal.
To prove (4.1), let C˜ ′ → C ′ be the minimal desingularisation of C ′. According to
Lemma 4.1.2, we have
H0(Emin, ωEmin/S) = H
0(C˜ ′, ωC˜′/S) ⊆ H0(C ′, ωC′/S).
The fact that C is obtained from Emin by contraction implies that f∗ωEmin/S = ωC/S,
see Lemma 4.1.8. Therefore, we have H0(C, ωC/S) = H0(Emin, ωEmin/S), and (4.1) holds.
2
4.2 Constructing minimal degree-n-models
Let φ be a genus one equation of degree n ∈ {1, 2, 3, 4} defining a smooth genus one
curve C/K. Assume that C(K) 6= ∅. Let E be the Jacobian elliptic curve of C with
minimal proper regular model Emin. If P ∈ C(K), then {P} will denote the Zariski
closure of {P} in Emin.
When n = 1, set D1 = 3.{P} where P ∈ C(K).When n ≥ 2, let
∑
(Pi) ∈ Div(C) be
aK-rational divisor of a hyperplane section on C. In particular, the degree of this divisor
is n. Assume moreover that {Pi} ∩ Emink is contained in one and only one irreducible
component of Emink . Consider the following Weil divisor on E
min
Dn =
∑
{Pi}.
Since Emin is regular, the divisor Dn is a Cartier divisor, see Remark 2.2.9.
36
We define an S-model Cn for C as follows
Cn := Proj(
∞⊕
m=0
H0(Emin,OEmin(mDn))).
There is a canonical morphism u : Emin −→ Cn contracting all the irreducible compo-
nents of Emink apart from the ones having nonempty intersection with Dn, see Theorem
2.2.11.
Lemma 4.2.1. Let Dn, n ∈ {1, 2, 3, 4}, be as above. Then H0(Emin,OEmin(mDn)),m ≥
1, is a free OK-module of rank 3m if n = 1, and of rank mn if n ≥ 2.
Proof: It is known that H0(Emin,OEmin(mDn)) ⊗OK K ∼= H0(C,OC(mDn|C)), see
for example ([20], Corollary 5.2.27). Moreover, by virtue of Riemann-Roch Theorem,
H0(C,OC(mDn|C)) is a 3m-dimensional K-vector space when n = 1, and an mn-
dimensional K-vector space when n ≥ 2.
SinceOEmin(mDn) is an invertible sheaf on Emin, it follows thatH0(Emin,OEmin(mDn))
is a flat OK-module, see ([20], Lemma 5.2.31). But since OK is a principal ideal
domain, an OK-module is flat if and only if it is torsion-free over OK . Therefore,
H0(Emin,OEmin(mDn)) is torsion-free over OK .
Since OK is a local ring, it is a general fact that a finitely generated flat OK-module
is free. Hence H0(Emin,OEmin(mDn)) is free over OK . Thus H0(Emin,OEmin(mDn)) is
a free OK-module of rank 3m if n = 1, and of rank mn if n ≥ 2. 2
Lemma 4.2.2. Let E be an elliptic curve over K with minimal proper regular model
pi : Emin → S. Let Dn be the divisor on Emin defined above. Then the following are true.
(i) H1(Emin,OEmin) is a free OK-module of rank 1.
(ii) For any m ≥ 2, there exists an exact sequence
0→ H0(Emin,OEmin((m− 1)Dn))→ H0(Emin,OEmin(mDn))→ A⊗m → 0,
where A is a free OK-module.
Proof: (i) Since OEmin is invertible on Emin, we have H1(Emin,OEmin) is a finitely
generated OK-module, see ([20], Theorem 5.3.2). Moreover, since OK is a local ring, it
follows that H1(Emin,OEmin) is free.
We have H1(Emin,OEmin)⊗OK K = H1(E,OE) ∼= K and H1(Emin,OEmin)⊗OK k =
H1(Emink ,OEmink ) ∼= k, see ([20], Lemma 9.4.28).
37
Let L := R1pi∗OEmin be the first higher direct image of OEmin . By definition we have
H0(S,L) = H1(Emin,OEmin), see ([20], Proposition 5.2.28). Now we have L is locally
free of rank 1 and hence H0(S,L) is a flat OK-module, see ([20], Lemma 5.2.31), which
implies that it is torsion-free over OK . It follows that H1(Emin,OEmin) is free of rank 1.
(ii) Let i : Dn → Emin be the canonical closed immersion. For any m ≥ 1, we
have a canonical isomorphism OEmin(mDn)⊗OEmin i∗ODn → i∗i∗OEmin(mDn), see ([20],
Exercise 5.1.1). But ODn fits in
0→ OEmin(−Dn)→ OEmin → ODn → 0.
Therefore, i∗i∗OEmin(mDn) ∼= OEmin(mDn) ⊗OEmin OEmin/OEmin(−Dn), and thus we
have the following short exact sequence
0→ OEmin((m− 1)Dn)→ OEmin(mDn)→ i∗i∗OEmin(mDn)→ 0. (4.2)
Consider the morphism Emin → PdS determined by Dn. We note that
(pii)∗(i∗OEmin(mDn)) ∼= (pii)∗(i∗OPdS(m)|Emin) ∼= (pii)∗(ODn(m)) ∼= ((pii)∗ODn)(m),
see ([20], Lemma 7.1.29 (b) and Exercise 5.1.16 (c)) for the second and last isomorphisms
respectively. Thus we have the following exact sequence by applying pi∗ to sequence (4.2)
0→ pi∗OEmin((m− 1)Dn)→ pi∗OEmin(mDn)→ L′⊗m → 0, (4.3)
where L′ = ((pii)∗ODn)(1). Taking global sections in (4.3) we have
0 → H0(Emin,OEmin((m− 1)Dn))→ H0(Emin,OEmin(mDn))→ H0(S,L′⊗m)
→ H1(Emin,OEmin((m− 1)Dn))→ . . . (4.4)
For m ≥ 1, we have H1(Emin,OEmin(mDn)) ⊗ K = H1(E,OE(mDn|K)) = 0, see for
example ([24], Chapter III). Therefore, we have H1(Emin,OEmin(mDn)) = 0 when m ≥
1. Therefore, taking m ≥ 2 in sequence (4.4) we have
0→ H0(Emin,OEmin((m− 1)Dn))→ H0(Emin,OEmin(mDn))→ H0(S,L′⊗m)→ 0.
Taking m = 1 in sequence (4.4) we will have
0→ OK → H0(Emin,OEmin(Dn))→ H0(S,L′)→ H1(Emin,OEmin)→ 0. (4.5)
Now L′ is an invertible sheaf over OS, therefore H0(S,L′) is a finitely generated torsion-
free OK-module over a local ring, and hence it is free. 2
The following theorem is classical for n = 1, 2, see for example ([20], §9.4) and [18].
Recall that when n = 2, we assume that char k 6= 2.
38
Theorem 4.2.3. Let Cn and Dn, n ∈ {1, 2, 3, 4}, be as above. Then there exists an
integral genus one equation φn of degree n defining Cn. Moreover, φn is minimal.
Proof: For n = 1, Lemma 4.2.2 (ii) allows us to pick a basis {1, x, y} of the free module
H0(Emin,OEmin(3D1)) such that {1, x} is a basis ofH0(Emin,OEmin(2D1)). Now proceed
as in §2.1 to write a genus one equation φ1 : f(x, y) = 0 of degree 1, where f is some
dependence relation in H0(Emin,OEmin(6D1)). The morphism
λ1 : C1 := Proj(
⊕
m≥0
H0(Emin,OEmin(mD1))) −→ P2S
associated to the basis {1, x, y} of H0(Emin,OEmin(3D1)) sends C1 into the cubic C ′1
defined by f(x, y) = 0. We know that both C1 and C ′1 are normal and integral, hence
λ1 : C1 → C ′1 is a birational morphism, see ([20], Exercise 3.2.6). Since the special fibers
of both C1 and C ′1 are irreducible, it follows that λ1 : C1 → C ′1 is an isomorphism, see
([20], Exercise 8.3.8 (b)).
For n = 2, we pick a basis {1, x} of H0(Emin,OEmin(D2)). Let λ2 be the morphism
λ2 : C2 := Proj(
⊕
m≥0
H0(Emin,OEmin(mD2)))→ P1S = SpecOK [x] ∪ SpecOK [1/x]
associated to the basis {1, x}. Let U = λ−12 (SpecOK [x]), V = λ−12 (SpecOK [1/x]). We
have C2 = U ∪ V. Taking the integral closure of OK [x] in K(C2), we have
OC2(U) = OK [x]⊕ yOK [x], for some y ∈ OC2(U),
moreover there exist g(x), f(x) ∈ OK [x] such that deg g ≤ 2, deg f ≤ 4 and y2+g(x)y =
f(x), see ([18], Lemme 1). As 2 is invertible in OK , we can complete the square and
assume that g = 0. The surjective homomorphism
OK [x, y]/(y2 − f(x))→ OC2(U), y 7→ y,
is an isomorphism because the left-hand term is integral of dimension 2, see ([20], Remark
8.3.25). Following the same argument we have OK [w, z]/(z2 − h(w)) ∼= OC2(V ), where
w = 1/x, z = y/x2, and h(w) = w4f(1/w). Therefore, C2 is the union of the two affine
open schemes
U = SpecOK [x, y]/(y2 − f(x)), V = SpecOK [w, z]/(z2 − w4f(1/w)).
For n = 3, 4, we pick a basis {x1, . . . , xn} of H0(Emin,OEmin(Dn)). Let λn : Emin →
Pn−1S be the morphism associated to the basis {x1, . . . , xn}. Let Zn be the closed subset
39
λn(E
min) ⊂ Pn−1S endowed with the reduced scheme structure. We are going to show
that Zn is defined by an integral genus one equation of degree n. Then we show that
Cn ∼= Zn, where Cn := Proj(
⊕
m≥0H
0(Emin,OEmin(mDn))).
When n = 3, the free OK-module H0(Emin,OEmin(3D3)) is of rank-9, see Lemma
4.2.1, but it contains the 10 elements x31, x
3
2, x
2
3, x
2
1x2, x
2
1x3, x
2
2x1, x
2
2x3, x
2
3x1, x
2
3x2, x1x2x3.
It follows that there are ai ∈ OK such that
F := a1x
3
1+a2x
3
2+a3x
3
3+a4x
2
1x2+a5x
2
1x3+a6x
2
2x1+a7x
2
2x3+a8x
2
3x1+a9x
2
3x2+a10x1x2x3 = 0.
Rescaling x, y and z, we can assume that there is at least one ai ∈ O∗K . Hence Z3 is
contained in ProjOK [x1, x2, x3]/(F ).
When n = 4, we consider the 10 elements x21, x1x2, x1x3, x1x4, x
2
2, x2x3, x2x4, x
2
3, x3x4, x
2
4
in the rank-8 free OK-module H0(Emin,OEmin(2D4)). They satisfy two linearly inde-
pendent quadrics Q and R. Therefore, Z4 is contained in the intersection of Q and
R.
We want to show that Zn = ProjOK [x1, . . . , xn]/In, where I3 = (F ) and I4 =
(Q,R). Since Zn ⊆ ProjOK [x1, . . . , xn]/In, we have ProjOK [x1, . . . , xn]/In = Zn ∪
Z ′n, for some closed subscheme Z
′
n ⊂ Pn−1S , Z ′n 6= ProjOK [x1, . . . , xn]/In. Recall that
C is the generic fiber of Emin. Since Dn|C is a divisor of degree n, n ∈ {3, 4}, on
C, it follows that Proj(OK [x1, . . . , xn]/In ⊗ K) is irreducible, see ([24], Chapter III).
Hence, ProjOK [x1, . . . , xn]/In is irreducible itself, see for example ([2], Lemma 2.2). It
follows from the definition of irreducibility that Z ′n = ∅, and the closed subscheme Zn
is ProjOK [x1, . . . , xn]/In.
According to the description of the contraction morphism included in the proof of
([20], Proposition 8.3.30), the morphism λn : E
min → Zn ⊆ Pn−1S , n = 3, 4, factors into
un : E
min → Cn followed by vn : Cn → Zn, where vn is the normalisation morphism. It is
understood that vn is a finite morphism, hence for an irreducible component Γ of E
min
k ,
λn(Γ) is a point if and only if un(Γ) is a point. In other words, the special fibers of Cn
and Zn have the same number of irreducible components. We have shown that Zn is
integral of dimension 2. Hence both Cn and Zn have dimension 2, their generic fibers are
isomorphic, and their special fibers have the same number of irreducible components.
By virtue of ([20], Remark 8.3.25), vn : Cn → Zn is an isomorphism.
Since Cn is obtained by contracting components in Emin, i.e., the minimal desingu-
larisation of Cn is isomorphic to Emin, we have that Cn is minimal, see Proposition 4.1.9.
2
40
4.3 Geometric criteria
In this section we will prove Theorem 4.0.1 and state some direct corollaries.
Lemma 4.3.1. Let φ be an integral genus one equation of degree n = 1, 2, 3, 4. As-
sume that φ defines a smooth genus one curve C over K, and that C(K) 6= ∅. Assume
moreover that φ defines a normal degree-n-model C for C. Let E/K be the Jacobian
elliptic curve of C, and Emin be the minimal proper regular model of E. Then we have
H0(Emin, ωEmin/S) = H
0(C, ωC/S) as sub-OK-modules of H0(E, ωE/K) if and only if φ is
minimal.
Proof: Assume that H0(Emin, ωEmin/S) = H
0(C, ωC/S). Let C ′ be another degree-n-
model for C. Let C˜ ′ → C ′ be the minimal desingularisation of C ′. Then we have
H0(C, ωC/S) = H0(Emin, ωEmin/S) = H0(C˜ ′, ωC˜′/S) ⊆ H0(C ′, ωC′/S),
see Lemma 4.1.2. Therefore, φ is minimal by virtue of Lemma 4.1.7.
Now assume that φ is minimal. Let H be the K-rational hyperplane section divisor
defined by φ, see §2.1. If n = 1, then H = 3(P ) for some P ∈ C(K). Set D1 = 3{P},
where {P} is the Zariski closure of {P} in Emin. If n ≥ 2, then pick x ∈ Emink such that
x lies on a multiplicity-1 component and on no other component, and x is defined over
k. Hensel’s Lemma allows us to lift x to a point P ∈ C(K). Set Q ∈ C(K) to be such
that (Q) ∼ H − (n− 1).(P ). Set Dn = (n− 1).{P}+ {Q}.
Consider the S-model C ′ for C given by
C ′ = Proj(
∞⊕
m=0
H0(Emin,OEmin(mDn))).
Let φ′ be the minimal genus one equation of degree n defining C ′, see Theorem 4.2.3.
Since Hn and Dn|C have the same degree and sum, they are linearly equivalent and the
genus one equations φ and φ′ are K-equivalent, see for example [10].
Moreover, since C ′ is obtained from Emin by contraction, Lemma 4.1.8 shows that
ωEmin/S = ω
′OEmin where ω′ ∈ H0(E, ωE/K) is such that ωC′/S = ω′OC′ .
Since the genus one equations φ and φ′ are both minimal, in particular they have
the same level, Lemma 4.1.7 implies the second equality of the following
H0(Emin, ωEmin/S) = H
0(C ′, ωC′/S) = H0(C, ωC/S).
2
41
Proof of Theorem 4.0.1: We proved one of the implications of the theorem in
Proposition 4.1.9.
Assume that φ is minimal and that ωC/S = ωOC for some ω ∈ H0(E, ωE/K).
We assume on the contrary that C˜ 6∼= Emin, and therefore Ck contains an exceptional
divisor Γ.
Let B be the set of points x ∈ C˜ where ωC˜/S is not generated by its global sections.
Since Γ is an exceptional divisor, we have deg ωC˜/S|Γ < 0, see ([20], Proposition 9.3.10),
it follows that H0(Γ, ωC˜/S|Γ) = 0, therefore Γ ⊆ B. But we have
ωC˜/S|Γ = ωC/S|Γ = ωOC|Γ,
and the global sections of ωC˜/S are
H0(C˜, ωC˜/S) = H0(Emin, ωEmin/S) = H0(C, ωC/S) = ωOK ,
where the second equality is justified by C being minimal, see Lemma 4.3.1. Therefore,
ωC˜/S is generated by its global sections at every x ∈ Γ, whence a contradiction. Thus
Ck contains no exceptional divisors and C˜ ∼= Emin. 2
Now we state more criteria for normal degree-n-models to be minimal.
Corollary 4.3.2. Let φ,C, C, E, C˜ and Emin be as in Theorem 4.0.1. Assume that
ωC/S = ωOC for some ω ∈ H0(E, ωE/K). Then the following statements are equivalent.
(i) φ is minimal.
(ii) ωC˜/S = ωOC˜.
(iii) H0(C˜, ωC˜/S) = H0(C, ωC/S) = ωOK .
Proof: (i)⇒ (ii): Since φ is minimal, we have C˜ ∼= Emin. But ωEmin/S = ωOEmin , see
Lemma 4.1.8. Whence (ii).
(ii)⇒ (iii) follows directly from Lemma 4.1.6.
(iii) ⇒ (i): Since H0(C˜, ωC˜/S) = H0(Emin, ωEmin/S), see Lemma 4.1.2, then φ is
minimal, see Lemma 4.3.1. 2
In §3.2 we stated all the combinatorial possibilities for the special fiber of a degree-
n-model C for a smooth genus one curve C. Theorem 4.0.1 allows us to find out which
of these possibilities occur for minimal degree-n-models for C according to the Kodaira
symbol of the Jacobian elliptic curve E of C. That can be done by looking at the pull-
back of the irreducible components of Ck \ Sing(C) under the contraction morphism
42
u : Emin → C, where Sing(C) is the singular locus of C. The pull-back of an irreducible
component Γ of Ck \ Sing(C) in Emink is called the strict transform of Γ. It is understood
that the strict transform of an irreducible component Γ has the same multiplicity as Γ.
For example, if Ck consists of a conic and a double line, then the strict transform of Ck
in Emink consists of a multiplicity-1 irreducible component corresponding to the conic,
and a multiplicity-2 irreducible component corresponding to the double line.
We set
T1 = {nodal cubic, conic + line, three lines},
T2 = {cuspidal cubic, conic + tangent, three concurrent lines},
T3 = {line + double line, triple line},
T4 =
 nodal quartic, secant conics, conic + two lines not crossing on it, four lines,cubic + secant line
 ,
T5 =
 cuspidal quartic, tangent conics, conic + two lines crossing on it,four concurrent lines, cubic + tangent line
 ,
T6 = {two lines + double line, conic + double line, double conic, two double lines},
T7 = {triple line + line, quadruple line}.
In the following corollary we assume that k is algebraically closed.
Corollary 4.3.3. Let φ be a minimal genus one equation of degree n = 1, 2, 3, 4. Assume
that φ defines a smooth genus one curve C, C(K) 6= ∅. Let E be the Jacobian elliptic
curve of C with Kodaira symbol T. Assume that C is a minimal degree-n-model for C
defined by φ. Then Ck lies in one of the sets determined by the following tables.
T n=1 n=2
I0 {smooth cubic} {smooth quartic}
Im,m ≥ 1 {nodal cubic} {nodal quartic, intersecting lines}
II {cuspidal cubic} {cuspidal quartic}
III, IV {cuspidal cubic} {cuspidal quartic, tangent conics}
I∗m,m ≥ 0, IV∗, III∗ {cuspidal cubic} {cuspidal quartic, tangent conics, double line }
II∗ {cuspidal cubic} {cuspidal quartic, double line}
43
T n=3 n=4
I0 {smooth cubic} {smooth quartic}
Im,m ≥ 1 T1 T4
II {cuspidal cubic} {cuspidal quartic}
III {cuspidal cubic, conic+tangent} {cubic+tangent}∪
{cuspidal quartic, tangent conics}
IV T2 T5 \ {four concurrent lines}
I∗m,m ≥ 0 T2∪{line+double line} T5 ∪ T6
IV∗ T2 ∪ T3 T6∪ {triple line+line}∪
T5\ {four concurrent lines}
III∗ T3∪ T6 ∪ T7∪{cuspidal quartic}∪
{cuspidal cubic, conic+tangent} {tangent conics, cubic+tangent}
II∗ T3∪{cuspidal cubic} {cuspidal quartic}∪T7∪
T6\{two lines+double line}
Proof: The case n = 1 is already known, see ([27], Chapter III, Proposition 1.4).
In order to classify which of these forms of the special fibers occur when E has
multiplicative reduction and which occur when E has additive reduction, we need to
compute the valuations of the invariants c4, c6, and ∆ corresponding to φ. For explicit
formulae for c4, c6, and ∆ see ([11], Lemma 2.9).
For n = 2, if Ck is either a nodal quartic or two intersecting lines, then ν(c4) = 0,
and ν(∆) ≥ 1, hence E has multiplicative reduction. Similarly, for n = 3, 4, if Ck lies
in T1, T4 respectively, then E has multiplicative reduction. The remaining forms of Ck
force ν(c4) ≥ 1, ν(∆) ≥ 1, therefore E has additive reduction.
Now since φ is minimal, it follows that Emin ∼= C˜, where C˜ → C is the minimal
desingularisation of C, see Theorem 4.0.1. The strict transforms of the irreducible com-
ponents of Ck are irreducible components in Emink with the same multiplicities. We
consider the multiplicities of the irreducible components of Ck, and the number lm of
irreducible components with multiplicity-m in Ck. If the graph associated to Emink has
components with the same multiplicities as those of Ck, and Emink contains at least lm
components with multiplicity-m, then the strict transform of Ck can lie in Emink , hence
we obtain the classification given above. For a reference for the graphs associated to
Emink see ([28], Chapter IV, Table 9.4.1). 2
44
Chapter 5
Isomorphisms of degree-n-models
For this chapter we assume that K is a Henselian discrete valuation field with ring of
integers OK . The residue field k is algebraically closed, t is a uniformiser, and S =
SpecOK .
In the first section of this chapter we see the conditions under which two minimal
degree-n-models are isomorphic. In fact, we show that two minimal degree-n-models
are isomorphic if and only if they have the same special fiber, see Theorem 5.1.4 below.
In the second section we assume that Γ is an irreducible component of multiplicity-
1 in Emink , then we count the number of minimal degree-n-models with special fibers
containing a multiplicity-1 irreducible component whose strict transform in Emink is Γ.
The results obtained in the second section will not be used elsewhere in this thesis.
5.1 Isomorphic degree-n-models
Let (C, α) be a minimal degree-n-model for a smooth genus one curve C over K. Assume
that C(K) 6= ∅. Let E be the Jacobian elliptic curve of C, and Emin be the minimal
proper regular model of E. Let P ∈ C(K). Fix an isomorphism β : C → E such that
β(P ) = 0E. The isomorphism β identifies the group structure on the elliptic curve (C,P )
with the group structure on (E, 0E), hence β extends to an isomorphism between the
minimal proper regular model of C and Emin.
Let C be a degree-n-model for a smooth genus one curve over K. Let Γ be an
irreducible component of Ck. We will write degk(Γ) for the degree of Γ. By the type
of Γ we mean the ordered pair (multk(Γ), degk(Γ)), where multk(Γ) is the multiplicity
of Γ defined in §3.2.
45
Remark 5.1.1. Let (C1, α1) and (C2, α2) be two isomorphic minimal degree-n-models
for a smooth genus one curve C over K. Assume that C(K) 6= ∅. Let E be the Jacobian
of C with minimal proper regular model Emin. Let α := α−12 α1 : (C1)K → (C2)K . By
the definition of isomorphic degree-n-models, the map α extends to an S-isomorphism
α˜ : C1 → C2 which is defined by an element in Gn(OK). Therefore, if Γ is an irreducible
component of (C2)k, then α˜∗Γ is an irreducible component of (C1)k with the same type as
Γ. Moreover, after identifying the minimal proper regular model of C with Emin, (C1)k
and (C2)k have the same strict transform in Emink .
In this section we show that the converse of Remark 5.1.1 holds. More precisely, if
the strict transforms of the special fibers of two minimal degree-n-models coincide in
the sense given in Theorem 5.1.4, then these degree-n-models are isomorphic.
Let x be a generator of K(P1K) over K. We define the S-scheme P1x to be
P1x := SpecOK [x] ∪ SpecOK [1/x].
We have P1x ∼= P1S as S-schemes and P1x is an S-model for P1K . Furthermore, every
smooth S-model for P1K is obtained that way, see ([18], §4). Two S-models P1x,P1u for
P1K are isomorphic if and only if there exists a matrix a c
b d
 ∈ GL2(OK)
such that x = (au+ b)/(cu+ d).
Now we use models of projective lines to describe isomorphic degree-2-models for
smooth genus one curves over K.
Theorem 5.1.2. Let (C1, α1) and (C2, α2) be two minimal degree-n-models, n = 1, 2, for
a smooth genus one curve C over K. Assume that C(K) 6= ∅. Set α : α−12 α1. Then the
following statements are equivalent.
(i) The map α extends to an isomorphism α˜ : C1 → C2 of S-schemes.
(ii) The curves (C1)k and (C2)k have the same strict transform in Emink .
(iii) (C1, α1) and (C2, α2) are isomorphic as degree-n-models for C.
Proof: This is clear for the case n = 1 as there is a unique minimal degree-1-model
for C. So we will assume that n = 2.
(i)⇔ (ii) : It is known that a birational map induces an S-isomorphism if and only if
it establishes a bijection between the generic points of the special fibers, see for example
46
([20], Remark 8.3.25). Hence (i) is equivalent to the statement that any irreducible
component of (C2)k is carried to an irreducible component of (C1)k under α˜∗. The latter
occurs if and only if the strict transforms of (C1)k and (C2)k coincide in Emink because α˜
extends to the identity on Emin.
(i)⇔ (iii) : The degree-2-model Ci, i = 1, 2, is obtained from Emin by a contraction
morphism, see Theorem 4.0.1. Let Di be a Cartier divisor on E
min such that
Ci := Proj(
⊕
m≥0
H0(Emin,OEmin(mDi))).
Let {1, xi} be a basis for the free module H0(Emin,OEmin(Di)). Now consider the mor-
phism Ci xi−→ P1xi , we have the commutative diagram
C1 x1 //
α˜
²²
P1x1
α˜
²²
C2 x2 // P1x2
Therefore, C1 and C2 are S-isomorphic if and only if P1x1 and P1x2 are S-isomorphic.
The latter occurs if and only if α˜ : P1x1 → P1x2 is defined by an element in GL2(OK)
which means that C1 and C2 are isomorphic degree-2-models for C. 2
The following example, shown to me by T. Fisher, illustrates that Theorem 5.1.2
does not hold for n = 4.
Example 5.1.3. Consider the following forms
F1 = tx
2
1 − x2x4 + x23, F2 = tx24 − x1x3 + x22,
F ′1 = x
2
1 − x2x4 + tx23, F ′2 = x24 − x1x3 + tx22.
The genus one equations φ : F1 = F2 = 0 and φ
′ : F ′1 = F
′
2 = 0 are minimal of degree
4. The genus one equation φ defines a smooth genus one curve C. Let (C, id4) and
(C ′, α′) be two minimal degree-4-models for C, where C is given by φ, C ′ is given by
φ′, and α′ is multiplying x2 and x3 by t and dividing through by t. Now Ck consists
of the line Γ1 : {x2 = x3 = 0} and the twisted cubic Γ2 parameterised by [u : v] 7→
[u3 : u2v : uv2 : v3]. Similarly, C ′k consists of Γ′1 : {x1 = x4 = 0} and the twisted cubic
Γ′2 : [u : v] 7→ [u2v : u3 : v3 : uv2]. The morphism α′ carries the generic point of Γ1 to the
generic point of Γ′2 and the generic point of Γ2 to the generic point of Γ
′
1. But (C, id4)
and (C ′, α′) are not isomorphic degree-4-models.
47
The above example shows that we need to modify Theorem 5.1.2 when n = 3, 4.
Let (Ci, αi), i = 1, 2, be a minimal degree-n-model for a smooth genus one curve
C → Pn−1K , n = 3, 4. Let Di be a divisor on Emin which defines Ci, more precisely Ci
is obtained from Emin by contraction using Di. What we are going to do next is to
compare the divisors D1|Emink and D2|Emink .
Recall that the degree map degk : Pic(P1k) → Z is an isomorphism. Moreover,
Pic0(P1k) = 1, see for example ([20], Proposition 9.3.16). Therefore, two effective divisors
D1, D2 on P1k are linearly equivalent if and only if they have the same degree.
Let X be a projective curve over k. Let X1, . . . , Xm be its irreducible components
with respective multiplicities d1, . . . , dm. We give each Xi the reduced subscheme struc-
ture. If L is an invertible sheaf on X, then we define the partial degree of L on Xi to be
deg(L|Xi). Then we have
degL =
m∑
i=1
di deg(L|Xi),
see for example ([20], Proposition 7.5.7) or ([5], §9.1, Proposition 5).
We consider the families of invertible sheaves of degree 0 on X, Pic0(X). Since k
is algebraically closed, Corollary 13 of ([5], §9.3) shows that Pic0(X) consists of all
elements of Pic(X) whose partial degree on each irreducible component Xi is zero.
Now we are in a place to generalise Theorem 5.1.2. Recall that the type of an
irreducible component Γ is the pair (multk(Γ), degk(Γ)).
Theorem 5.1.4. Let (C1, α1) and (C2, α2) be two minimal degree-n-models, n = 1, 2, 3, 4,
for a smooth genus one curve C over K. Assume that C(K) 6= ∅. Set α = α−12 α1 and
denote its extension C1 99K C2 by α˜. Then the following statements are equivalent.
(i) (C1, α1) and (C2, α2) are isomorphic as degree-n-models for C.
(ii) For every irreducible component Γ of (C2)k, α˜∗Γ is an irreducible component of
(C1)k with the same type as Γ.
Proof: The cases n = 1, 2 have been done in Theorem 5.1.2.
Let n = 3, 4. That (i) implies (ii) follows from Remark 5.1.1.
(ii)⇒ (i): We want to show that α˜ is defined by an element in Gn(OK).
Let Emin be the minimal proper regular model of the Jacobian E of C. Statement
(ii) implies that both (C1)k and (C2)k have the same strict transform in Emink .
Let Di be a defining divisor of Ci as a contraction in Emin, i.e.,
Ci = Proj(
⊕
m≥0
H0(Emin,OEmin(mDi))).
48
Set Li = OEmin(Di), Di,k = Di|Emink and Li,k = Li|Emink .
Let Γ be an irreducible component of (C2)k. Consider the strict transform Γ˜ of the
irreducible components Γ and α˜∗Γ in Emink . Since Γ and α˜
∗Γ have the same type, it
follows that degk L1,k|Γ˜ = degk L2,k|Γ˜. For any irreducible component Λ which is not a
strict transform of a component of (Ci)k, we have degk Li,k|Λ = 0. Since each irreducible
component of Emink is isomorphic to P1k and D1,k, D2,k have the same degree on each
irreducible component of Emink , we have D1,k ∼ D2,k and L1,k ∼= L2,k.
Now we know that C(K) 6= ∅, L1|(C1)K ∼= L2|(C2)K , and we have established the
isomorphism L1,k ∼= L2,k, these imply that L1 ∼= L2, see ([20], Exercise 9.1.13 (b)).
Therefore, H0(Emin,L1) and H0(Emin,L2) are isomorphic as OK-modules, and α˜ is a
change of basis of a free OK-module of rank n, hence α˜ is defined by an element in
Gn(OK). 2
In the following corollary we assume that char k 6= 2 when n = 2.
Corollary 5.1.5. Let C a smooth genus one curve. Assume that C(K) 6= ∅. Assume
moreover that the Jacobian E of C has either reduction types I0 or I1. Then there is a
unique minimal degree-n-model for C.
Proof: There is always a unique minimal degree-1-model for C. So assume n ≥ 2.
Let Emin be the minimal proper regular model of E. According to Theorem 4.0.1, any
minimal degree-n-model for C is obtained from Emin via contraction. Therefore, the
special fiber of a minimal degree-n-model for C consists of one irreducible component of
multiplicity-1 because Emink consists of a unique irreducible component of multiplicity-1.
By virtue of Theorem 5.1.4, if there are two minimal degree-n-models for C, then they
are isomorphic as they have the same strict transform in Emink . 2
5.2 Degree-n- and degree-(n− 1)-models, n ≥ 3
In this section we will count minimal degree-n-models with a specific property.
Let C be an S-model for a curve C/K. Let x be a closed point of Ck. Set
C+(x) := {P ∈ C : {P} ∩ Ck = {x}},
where {P} is the Zariski closure of {P} in C. C+(x) depends on the choice of the
model C. By K(P ) we mean the field of definition of P . We state the following lemma
which plays an essential rule in the way we construct divisors on minimal proper regular
models.
49
Proposition 5.2.1. Let C/K be a curve of genus g ≥ 1 with minimal proper regular
model Cmin. Fix a closed point x ∈ Cmink such that x lies on one and only one irreducible
component Γ of Cmink , of multiplicity r ≥ 1. Then there exists a point P ∈ C+(x) such
that [K(P ) : K] = r.
Proof: See ([20], Exercise 9.2.11 (c)) or ([19], Lemma 5.1 (b)). 2
Now we investigate finite field extensions of K.
Proposition 5.2.2. If [L : K] = m, where m ∈ {2, 3, 4}, then L = K( m√t), and L/K
is a Galois tamely ramified extension.
Proof: If a ∈ L, we will denote its image in k by a˜.
Since k = k, it follows that L is a totally ramified extension of K. The fact that
2, 3 - char(k) implies that this extension is tame. Let tL be a uniformiser for the ring of
integers OL of L. Then t = utmL for some u ∈ O∗L. Hensel’s Lemma implies that there is
a v ∈ O∗L such that vm = u.
Therefore, L = K[x]/(xm − t), which is clearly Galois for m = 2. For m = 3, 4, k
has a full set µm of m-th roots of unity, and we lift them to OK using Hensel’s Lemma.
Thus L is Galois. 2
Let C be a smooth genus one curve given by a minimal genus one equation of degree
n = 2, 3, 4 over K. Assume moreover that C(K) 6= ∅. Let E be the Jacobian elliptic
curve of C.
Let P ∈ C(K). Again we fix an isomorphism β : C → E such that β(P ) = 0E. So
we can dispense with C and write E instead. If D is an effective K-rational divisor of
degree n on E, then Riemann-Roch theorem implies that there exists a point Q ∈ E(K)
such that D ∼ (n − 1).0E + Q. Therefore, the equation defining C is an equation for
the double cover of the projective line E → P1K given by the divisor class [0E +Q] when
n = 2, or the image of E when it is embedded in Pn−1K by the divisor class [(n−1).0E+Q]
when n = 3, 4.
Consider the smooth genus one curve E
[(n−1).0E+Q]−−−−−−−−→ Pn−1K , n ≥ 2. Let Γ be an
irreducible component of Emink . We set
SΓ :=
 minimal degree-n-models (C, α) for E
[(n−1).0E+Q]−−−−−−−−→ Pn−1K such that Γ is
the strict transform of an irreducible component of Ck
 .
The following lemma shows that SΓ is not empty if the multiplicity of Γ is less than n.
Lemma 5.2.3. Let n ≥ 2. Let Γ be a multiplicity-m irreducible component of Emink . If
1 ≤ m < n, then SΓ 6= ∅.
50
Proof: Let x ∈ Emink be a closed point on Γ. Let P ∈ E+(x) be a closed point, i.e., P
is identified with its Galois orbit {P1, . . . , Pm}, where K(P ) = K(t1/m). Now set
Q2 = Q− (n− 1).P, D = (n− 1).{P}+ {Q2} if m = 1,
Q2 = Q−
∑n−1
i=1 Pi, D = {P}+ {Q2} if m = n− 1, n ≥ 3,
Q2 = Q− (P1 + P2), D = {P}+ {Q2}+ {0E} if m = 2, n = 4.
Now consider the following minimal degree-n-model for E
[(n−1).0E+Q]−−−−−−−−→ Pn−1K
C ′ = Proj(
⊕
m≥0
H0(Emin,OEmin(mD))),
see Theorem 4.0.1 and Theorem 4.2.3. We have x ∈ D∩Γ, and hence Γ is an irreducible
component of C ′k. Since D|E ∼ (n−1).0E+Q, it follows that there exists an isomorphism
α : C ′K ∼= E defined by an element in Gn(K). Thus (C ′, α) ∈ SΓ. 2
In what follows we compute the cardinality of SΓ when the multiplicity of Γ is 1.
Theorem 5.2.4. Consider the smooth genus one curve E
[(n−1).0E+Q]−−−−−−−−→ Pn−1K , n ≥ 3,
with a rational point on it. Let Γ be a multiplicity-1 irreducible component of Emink .
Let x ∈ Γ and P ∈ E+(x). Then there is a bijection between the set SΓ and the set of
minimal degree-(n− 1)-models for E [(n−2).0E+(Q−P )]−−−−−−−−−−−→ Pn−2K .
To prove Theorem 5.2.4 we have to establish a bijection between the two sets in
the statement. Let (C, α) ∈ SΓ. Let D be a defining divisor of C as a contraction in
Emin. The restriction D|E satisfies D|E ∼ (n − 1).0E + Q. Let P1 ∈ SuppD|E be such
that {P1} ∩Emink ∈ Γ. After applying a transformation in Gn(OK), we can assume that
P1 = P.
Now the bijection map is
λΓ : SΓ → {degree-(n− 1)-models for E [(n−2).0E+(Q−P )]−−−−−−−−−−−→ Pn−2K }
(C, α) 7→
(
Proj(
⊕
m≥0
H0(Emin,OEmin(mD′))), β
)
,
where D′ = D − {P}, and β is obtained from the linear equivalence D|E − (P ) ∼
(n − 2).0E + (Q − P ). The model on the right is a minimal degree-(n − 1)-model for
E
[(n−2).0E+(Q−P )]−−−−−−−−−−−→ Pn−2K , see Theorem 4.0.1 and Theorem 4.2.3.
Proof of Theorem 5.2.4: The map λΓ is well defined: Assume that (Ci, αi), i =
1, 2, are two isomorphic minimal degree-n-models in SΓ. Remark 5.1.1 shows that the
51
special fibers of both models have the same irreducible components with the same types,
therefore the irreducible components of the special fibers of λΓ((Ci, αi)), i = 1, 2, coincide
and have the same types. Thus λΓ((C1, α1)) and λΓ((C2, α2)) are isomorphic as degree-
(n− 1)-models for E [(n−2).0E+(Q−P )]−−−−−−−−−−−→ Pn−2K , see Theorem 5.1.4.
To prove that the map λΓ is injective: Assume that the images of (Ci, αi) ∈ SΓ, i =
1, 2, under λΓ are isomorphic degree-(n−1)-models, hence the special fibers of λΓ((Ci, αi))
consist of the same irreducible components with the same types. It follows that the
special fibers of both C1 and C2 have the same irreducible components with the same
types, because they both have the same irreducible components as the special fiber of
λΓ((Ci, αi)) with the degree of Γ being increased by 1. Therefore, Theorem 5.1.4 implies
that (C1, α1) and (C2, α2) are isomorphic.
Now we prove that λΓ is surjective. Let (C, α) be a degree-(n − 1)-model for
E
[(n−2).0E+(Q−P )]−−−−−−−−−−−→ Pn−2K with D being a divisor defining C as a contraction in Emin.
Consider the model (C ′, α′) ∈ SΓ defined by
Proj(
⊕
m≥0
H0(Emin,OEmin(m(D + {P})))),
and α′ is obtained from the linear equivalence D|E+(P ) ∼ (n−1).0E+(Q). The model
C ′ is defined by a minimal genus one equation of degree n, see Theorem 4.2.3. The
surjectivity follows by noting that λΓ((C ′, α′)) = (C, α). 2
52
Chapter 6
Computing in Emin
In this chapter K will be a perfect Henselian discrete valuation field with algebraic
closure K, ring of integers OK , normalised valuation ν, and a uniformiser t. We assume
that the residue field k is algebraically closed with char k 6= 2, 3.
Let E be an elliptic curve over K, with minimal proper regular model Emin. Let
E0(K) be the group of rational points of E with non-singular reduction. In this chapter
we define a map δm : Φ
m
K(E) → ΦK(E), where ΦmK(E) is the set of multiplicity-m
irreducible components of Emink , and ΦK(E) := Φ
1
K(E) is the group of components
E(K)/E0(K).
Since k is algebraically closed, it follows that the reduction of E is always split.
6.1 The components group
In this section we will study the components group ΦK(E) = E(K)/E
0(K). We start
by describing this group.
Proposition 6.1.1 ([28], Chapter IV, Corollary 9.2). Let E be an elliptic curve over K.
Let j(E) be the j-invariant of E. Then the group E(K)/E0(K) is finite. More precisely,
if E has multiplicative reduction, then E(K)/E0(K) is a cyclic group of order −ν(j(E));
otherwise, E(K)/E0(K) has order 1, 2, 3, or 4.
Assume E has reduction of type In, n ≥ 0. Then Emink is a non-singular smooth
curve of genus one when n = 0. Emink is a rational curve with a node when n = 1. E
min
k
consists of n multiplicity-1 irreducible components arranged in the shape of an n-gon
when n ≥ 2.
We will denote the extension of the normalised valuation ν on K to K by νK : K →
Q∪{∞}. Let p = char k. Denote by | ∗ | = 1/pνK(∗) an associated absolute value. Tate’s
53
uniformisation of E, see ([28], Chapter V, §3), implies that there exists a q ∈ K∗ with
ν(q) = ν(∆) = n such that
K
∗
/qZ ∼= E(K).
For z ∈ K∗, we denote by z˜ the image of z in E(K). For z ∈ K∗, there exists an integer
i such that |t|i+1 < |z| ≤ |t|i.
Consider the reduction map r : E(K) → E(k). Any isomorphism Z/nZ ∼−→ ΦK(E)
enables us to number the irreducible components Γi’s of E
min
k , with i¯ ∈ Z/nZ where i¯
denotes the class of the integer i in Z/nZ. There exists such a numbering such that if
|z| = |t|i, then r(z˜) belongs to the i-th component Γi, and if |t|i+1 < |z| < |t|i, then
r(z˜) is an intersection point of the irreducible components Γi and Γi+1. This intersection
point is unique when n ≥ 3. For the above discussion see ([21], p. 503).
Assume that E has additive reduction. We fix an isomorphism α : ΦK(E) ∼=
Z/nZ, n = 1, 2, 3, 4, or Z/2Z × Z/2Z when E has reduction of type I∗2m, m ≥ 0. Let Λ
be a multiplicity-1 irreducible component of Emink . By Hensel’s Lemma we can lift any
closed point x of Λ which lies on no other component to a rational point P ∈ E(K).
Now consider the image i¯ of P under α. We give the component Λ the number i, and
denote it by Γi.
The map δ1 : Φ
1
K(E)→ ΦK(E) will be viewed as the identification of multiplicity-1
components with their images in Z/nZ when the reduction is of type In, and with their
images under the isomorphism α defined above when the reduction is additive. In other
words, δ1(Γi) = i¯.
We note that if E has one of the reduction types In, n ≥ 0, II, III, or IV, then Emink
consists only of multiplicity-1 components, hence ΦmK(E) = ∅ when m ≥ 2 and we do
not need to define δm for m ≥ 2.
6.2 The set ΦmK(E), m ≥ 2
In this section we assume that E has one of the reduction types I∗m,m ≥ 0, IV∗, III∗, or
II∗. Then the special fiber of the minimal proper regular model Emin contains irreducible
components of multiplicities greater than one. In other words, there exists an integer
m ≥ 2 such that ΦmK(E) 6= ∅.
We define the map δm : Φ
m
K(E) → ΦK(E), m ≥ 2, as follows: Let x be a k-point
of Θ ∈ ΦmK(E). Recall that k is algebraically closed and hence x is a closed point.
Assume moreover that x lies on no other component. Let P ∈ E(K) be a point which
reduces to x such that [K(P ) : K] = m, see Proposition 5.2.1. Let σ be a generator of
Gal(K(P )/K). The sum of the Galois orbit sumP := P + . . . + P σ
m−1
of P is a point
54
in E(K). The image δm(Θ) is the image of sumP in ΦK(E) = E(K)/E
0(K). It will be
clear from the description of the map δm that δm(Θ) does not depend on x nor on P.
Since K(P )/K is a totally tamely ramified extension, see Proposition 5.2.2, the Ga-
lois group Gal(K(P )/K) is the inertia group of K(P )/K. In other words, Gal(K(P )/K)
fixes every irreducible component of Emink .
In what follows we are going to follow two strategies to determine the image of
each Θ ∈ ΦmK(E) under δm. When the reduction type is I∗n, we use Tate’s algorithm
to write down explicit equations for the components in Φ2K(E). For reduction types
IV∗, III∗ and II∗, the defining equations of the components in ΦmK(E), m ≥ 2, are
complicated. Therefore, we use the projection formula to exploit the symmetry of the
graphs associated to Emink .
6.2.1 Reduction type I∗n, n ≥ 0
Assume that E/K has reduction of type I∗n, n ≥ 0. The special fiber Emink contains
a sequence of multiplicity-2 components and no components of higher multiplicities.
Therefore, ΦmK(E) = ∅ when m ≥ 3. The strategy we will follow to compute δ2 in this
case is as follows: We desingularise a minimal Weierstrass model for E using a sequence
of blow-ups, then we determine conditions for points in E defined over K(
√
t) to lie on
one of the produced multiplicity-2 irreducible components, and use that to compute the
sums of the Galois orbits of these points.
We will use the data given by the proof of Tate’s algorithm to write the defining
equations of Θ ∈ Φ2K(E), see ([28], Chapter IV, §9). We start with a homogenised
minimal genus one equation of degree 1, i.e., a minimal Weierstrass equation of the
form
E : y2z + a1xyz + a3yz
2 = x3 + a2x
2z + a4xz
2 + a6z
3, ai ∈ OK ,
for E/K. We will write ai,r for t
−rai, and xr, yr for t−rx, t−ry respectively.
The proof of Tate’s algorithm shows that if E has additive reduction of one of the
types I∗n, n ≥ 0, IV∗, III∗ or II∗, then we can assume t | a1, a2, t2 | a3, a4, and t3 | a6.
Now blowing-up the singularity t = x = y = 0, by making the substitutions x = tx1 and
y = ty1 we will have
V : y21z + ta1,1x1y1z + ta3,2y1z
2 = tx31 + ta2,1x
2
1z + ta4,2x1z
2 + ta6,3z
3,
and its special fiber y21z = 0 consists of the double line y
2
1 = 0 and the multiplicity-1 line
z = 0. We next blow-up the line t = y1 = 0. Putting z = 1 and recalling the definitions
of xi and yi above, we obtain
V0 : ty
2
2 + ta1,1x1y2 + ta3,2y2 = x
3
1 + a2,1x
2
1 + a4,2x1 + a6,3,
55
and the special fiber of the total blow-up consists of
V˜ : y21z = 0, V˜0 : x
3
1 + a˜2,1x
2
1 + a˜4,2x1 + a˜6,3 = 0.
For type I∗0, the special fiber V˜0 consists of three distinct lines. Hence E
min
k consists of
the double line y21 = 0 together with four distinct lines of multiplicity 1 intersecting it.
Now we assume that h(x) = x3 + a˜2,1x
2 + a˜4,2x+ a˜6,3 has one double root. We may
assume that this root is x = 0, which implies that t2 - a2, t3 | a4, and t4 | a6. Now we
have
V˜0 : x
2
1(x1 + a˜2,1) = 0,
so we blow up the double line t = x1 = 0. Therefore, we make the substitution x1 = tx2
and divide by t to get
V1 : y
2
2 + ta1,1x2y2 + a3,2y2 = t
2x32 + ta2,1x
2
2 + ta4,3x2 + a6,4.
The total special fiber consists now of the simple lines z = 0 and x1 + a˜2,1 = 0, the
double lines y21 = 0 and x
2
1 = 0, and the special fiber of V1 which is given by
V˜1 : y
2
2 + a˜3,2y2 − a˜6,4 = 0.
If this quadratic equation has distinct roots in k, then V˜1 consists of two distinct simple
lines, and the reduction of E is of type I∗1. Otherwise, it has a double root and after a
translation on y2 we can take this double root to y2 = 0, i.e., t
3 | a3 and t5 | a6, and the
special fiber V˜1 of V1 is the double line y
2
2 = 0. We blow-up V1 along this double line via
the substitution y2 = ty3 and divide by t to get
V2 : ty
2
3 + ta1,1x2y3 + ta3,3y3 = tx
3
2 + a2,1x
2
2 + a4,3x2 + a6,5.
The special fiber is
V˜2 = a˜2,1x
2
2 + a˜4,3x2 + a˜6,5 = 0.
If this quadratic equation has distinct roots, then E has reduction type I∗2. Otherwise,
we continue the blowing-up process which will terminate eventually.
We note that at each step we have a new scheme Vl of the form
y2u + ta1,1xuyu + a3,uyu = t
ux3u + ta2,1x
2
u + ta4,u+1xu + a6,2u if l = 2u− 3 is odd,
ty2u+1 + ta1,1xuyu+1 + ta3,u+1yu+1 = t
u−1x3u + a2,1x
2
u + a4,u+1xu + a6,2u+1 if l = 2u− 2 is even.
The special fiber of Vl is given by
V˜l :
 y2u + a˜3,uyu − a˜6,2u = 0 if l = 2u− 3 is odd,a˜2,1x2u + a˜4,u+1xu + a˜6,2u+1 = 0 if l = 2u− 2 is even.
56
After each two blowing-ups the coefficients a3 and a4 are forced to be divisible by an
additional power of t. The special fiber V˜l consists of two distinct lines precisely when
l = n = ν(∆)− 6. Thus if E has reduction of type I∗n, n ≥ 0, we can assume that E has
coefficients with valuations as in the following table.
Table 6.1:
a1 a2 a3 a4 a6
n = 0 ≥ 1 ≥ 1 ≥ 2 ≥ 2 ≥ 3
n ≥ 2 even ≥ 1 = 1 ≥ n
2
+ 2 = n
2
+ 2 ≥ n+ 3
n odd ≥ 1 = 1 = n−1
2
+ 2 ≥ n−1
2
+ 2 ≥ n+ 3
The inequalities in the second, third, and fourth line relate to the valuation of the
coefficient in the first line.
The special fiber Emink is as in the following figure. The numbers on the components
refer to the multiplicities.
2
1 1
2
−−−−−
2
11
2
The number of multiplicity-2 components in Emink is n + 1. Let L = K(
√
t) be
the unique quadratic tame Galois extension of K. A point lying above a k-point on V˜l,
where l = 2u − 2 is even, and on no other component is a point in E(L) of the form
(αtu+1 + βtu+1/2, y), where y ∈ L, tu+1 | y and α, β ∈ OK with ν(α) ≥ ν(β) = 0. We
have β ∈ O∗K , since otherwise this point would reduce to a point on V˜l+1.
A point lying above a k-point on V˜l, where l = 2u − 3 is odd, and on no other
component is a point in E(L) of the form (x, αtu+1 + βtu+1/2), where x ∈ L, tu | x and
α, β ∈ OK with ν(α) ≥ ν(β) = 0. We have β ∈ O∗K , since otherwise this point would
reduce to a point on V˜l+1.
We recall the addition formula on elliptic curves. Let P1, P2 ∈ E(K). If P1 6= ±P2,
then
x(P1+P2) = λ
2+a1λ−a2−x(P1)−x(P2), and, y(P1+P2) = −(λ+a1)x(P1+P2)−µ−a3,
57
where
λ =
y(P2)− y(P1)
x(P2)− x(P1) , µ =
y(P1)x(P2)− y(P2)x(P1)
x(P2)− x(P1) .
Lemma 6.2.1. Assume E has one of the reduction types I∗n, n ≥ 0, IV∗, III∗ or II∗.
Let L = K(
√
t) with Gal(L/K) = 〈σ〉. Let P ∈ E(L) be a point which lies above the the
multiplicity-2 component Θ0 : y
2
1 = 0. Then P +P
σ ∈ E0(K). In particular, δ2(Θ0) = 0¯.
Proof: We can assume that P = (x, αt2 + βt3/2), where x ∈ L, t | x and α, β ∈ OK ,
with ν(α) ≥ ν(β) = 0. If P σ = −P, then P + P σ = 0 ∈ E0(K) and we are done. Since
t | x(P ), we assume x(P ) = α′t + β′t3/2, where α′, β′ ∈ OK . Therefore, we have in the
addition formula given above that
λ :=
y(P )− y(P σ)
x(P )− x(P σ) =
β
β′
∈ K.
If ν(β′) = 0, then ν(x(P + P σ)) = 0 because t | a1, a2, see Table 6.1. Hence P + P σ is
not a singular point, i.e., P +P σ ∈ E0(K). If ν(β′) > 0, then we have ν(x(P +P σ)) < 0,
in particular P + P σ ∈ E0(K), and we are done. 2
Now we state our main result of this subsection in the following proposition.
Proposition 6.2.2. Assume that E/K has reduction type I∗n, n ≥ 0. Let L = K(
√
t)
with Gal(L/K) = 〈σ〉.
(i) The multiplicity-1 component Γ given by x1 + a˜2,1 = 0 is of order 2 in ΦK(E).
(ii) Let P ∈ E(L) be a point lying above the double line y21 = 0. Then P+P σ ∈ E0(K).
(iii) Let P ∈ E(L) be a point lying above V˜l where l = 2u − 2 is even. Then P + P σ
reduces to a point on Γ.
(iv) Let P ∈ E(L) be a point lying above V˜l where l = 2u− 3 is odd. Then P + P σ ∈
E0(K).
Proof: We note first that the identity component, i.e., points in E0(K), is given by
the linear equation z = 0.
(i) If P ∈ E(K), then x(−P ) = x(P ). If moreover P lies above Γ : x1 + a˜2,1 = 0,
then −P lies above Γ as well. It follows that the inverse of Γ when considered as an
element of ΦK(E) is itself, i.e., Γ has order 2 as an element in ΦK(E).
(ii) This is Lemma 6.2.1.
(iii) We can assume that P = (αtu+1 + βtu+1/2, y), where y ∈ L, tu+1 | y and
α, β ∈ OK with ν(α) ≥ ν(β) = 0. Therefore, we have ν(λ) > 0 where λ := y(P )−y(Pσ)x(P )−x(Pσ) .
58
Following the addition formula we have x(P + P σ) = λ2 + a1λ − a2 − x(P ) − x(P σ),
dividing by t and using that t | a1, a2, see Table 6.1, we get that x1(P + P σ) = −a˜2,1
mod t, and we are done.
(iv) We can assume that P = (x, αtu+1+βtu+1/2), where x ∈ L, tu | x and α, β ∈ OK ,
with ν(α) ≥ ν(β) = 0. If P σ = −P, then P + P σ = 0 ∈ E0(K) and we are done. Thus
we assume x(P ) = α′tu + β′tu+1/2, where α′, β′ ∈ K with ν(α′) ≥ ν(β′) ≥ 0. Therefore,
we have in the addition formula given above that λ = β
β′ ∈ K. If ν(β′) = 0, then
ν(x(P + P σ)) = 0, hence P + P σ is not a singular point, i.e., P + P σ ∈ E0(K). If
ν(β′) > 0, then we have ν(x(P + P σ)) < 0, in particular P + P σ ∈ E0(K). 2
In Proposition 6.2.2 we note that since Γ : x1 + a˜2,1 = 0 has order 2, we have
that δ1(Γ) = 2¯ ∈ ΦK(E) ∼= Z/4Z, when n is odd. When n is even, we know that
α : ΦK(E) ∼= Z/2Z × Z/2Z, therefore every non-identity irreducible component has
order 2. From now on, we will fix α such that δ1(Γ) = (1¯, 1¯). We obtain the following
direct consequence.
Corollary 6.2.3. Assume that E/K has reduction type I∗n, n ≥ 0. Let
Φ2K(E) = {V˜−1 : y21 = 0, V˜0, . . . , V˜n−1}
be as above. Then δ2 : Φ
2
K(E)→ ΦK(E) is determined according to the following.
δ2(Θ) =

0¯ if Θ = V˜l, where l is odd and n is odd,
2¯ if Θ = V˜l, where l is even and n is odd,
(0¯, 0¯) if Θ = V˜l, where l is odd and n is even,
(1¯, 1¯) if Θ = V˜l, where l is even and n is even.
While treating elliptic curves with reduction type III∗, see below, we will need more
information on the reduction type I∗0. If E/K is an elliptic curve given by a minimal
Weierstrass equation with reduction type I∗0, then the special fiber of the minimal proper
regular model of E consists of the single line z = 0, a double line y21 = 0, and the three
distinct lines of
V˜0 : x
3
1 + a˜2,1x
2
1 + a˜4,2x1 + a˜6,3 = 0.
Using Hensel’s Lemma we can lift the simple zeros of this cubic polynomial to α1, α2,
and α3 in OK such that
x3 + a2x
2 + a4x+ a6 = (x− tα1)(x− tα2)(x− tα3).
Let Pi = (tαi, 0). We conclude that the reduction map {0, P1, P2, P3} → ΦK(E) is
surjective as each of these points lies on a different multiplicity-1 component. Note that
these points are the elements of E(K)[2].
59
6.2.2 Reduction types IV∗, III∗ and II∗
Let E/K be an elliptic curve with either reduction types IV∗ or III∗. We start by
investigating the new reduction type of E over Galois tame extensions.
Lemma 6.2.4. Let E/K be an elliptic curve. Let Lm = K(tm), where tm = t
1/m, m ∈
{2, 3, 4}. The following statements are true.
(i) If E/K has reduction type IV∗, then E/L3 has good reduction.
(ii) Assume that E/K has reduction type III∗. Then E/L2 has reduction type I∗0, E/L3
has reduction type III, and E/L4 has good reduction.
Proof: Let νLm denote the normalised discrete valuation on Lm.
(i) We recall from ([28], Chapter IV, §9) that if E has reduction type IV∗, then
ν(c4) ≥ 3, ν(c6) = 4, and ν(∆) = 8. Therefore, νL3(c4) ≥ 9, νL3(c6) = 12, and
νL3(∆) = 24. Since charK 6= 2, 3, a Weierstrass equation for E/L3 with good reduction
is now given by
y2 = x3 − 27t−83 c4x− 54t−123 c6.
(ii) Recall from [28] that if E has reduction type III∗, then ν(c4) = 3, ν(c6) ≥ 5, and
ν(∆) = 9. Therefore, νLm(c4) = 3m, νLm(c6) ≥ 5m, and νLm(∆) = 9m. The following
Weierstrass equation for E/Lm has reduction type I
∗
0 when m = 2, reduction type III
when m = 3, and good reduction when m = 4
y2 = x3 − 27t−4(m−1)m c4x− 54t−6(m−1)m c6.
2
Now we assume that E/K has additive reduction of type IV∗. The minimal proper
regular model Emin has special fiber as in the following figure. Again the numbers on
the components refer to the multiplicities.
3
Λ
2 Θ0
1
Γ0
2 Θ1
1
Γ1
2 Θ2
1
Γ2
60
According to the proof of Tate’s algorithm, the blowing-up process will yield that the
multiplicity-1 identity component Γ0 is given by the equation z = 0, and the multiplicity-
2 component Θ0 is given by the equation y
2
1 = 0. The multiplicity-3 component Λ has
equation x31 = 0, see ([28], Chapter IV, §9). The multiplicity-1 component Γi will
correspond as usual to i¯ ∈ ΦK(E).
Proposition 6.2.5. Assume that E/K has reduction type IV∗. Let Lm = K(tm), where
tm = t
1/m, m ∈ {2, 3}. Let Gal(Lm/K) = 〈σm〉.
(i) Let P ∈ E(L2) lie above a multiplicity-2 component Θi, i ∈ {0, 1, 2}. The following
table relates the component Θi to the multiplicity-1 component above which P+P
σ2
lies, and the image δ2(Θi).
P˜ ∈ Θ0 Θ1 Θ2
(P + P σ2)∼ ∈ Γ0 Γ2 Γ1
δ2 0¯ 2¯ 1¯
(ii) Let P ∈ E(L3) be a point lying above Λ. Then we have
∑2
i=0 P
σi3 ∈ E0(K). In
particular, δ3(Λ) = 0¯.
Proof: (i) If P ∈ E(L2) lies above Θ0 : y21 = 0, then P + P σ2 ∈ E0(K), see Lemma
6.2.1.
Let P ∈ E(L2) be a point lying above Θ1 and above no other component. Now
let Q ∈ E(K) be a point lying above Γ2. Consider the translation-by-Q automorphism
τQ : E → E. This K-automorphism extends to give an OK-automorphism τ ′Q : Emin →
Emin, see e.g. ([28], Chapter IV, Proposition 4.6). Moreover, τ ′∗Q (Γ1) = Γ0, and the
projection formula implies that τ ′∗Q (Γ1).τ
′∗
Q (Θ1) = Γ1.Θ1, see ([20], Theorem 9.2.12),
therefore τ ′∗Q (Θ1) = Θ0. Therefore, τQ(P ) lies above Θ0, hence the first part of the proof
shows that τQ(P )+ τQ(P )
σ2 ∈ E0(K), but τQ(P )σ2 = P σ2 +Q. Whence P +P σ2 +2Q ∈
E0(K), in other words P + P σ2 ∈ Q+ E0(K), i.e., P + P σ2 lies above Γ2.
If P ∈ E(L2) lies above Θ2 and above no other component, then we choose Q ∈ E(K)
to lie above Γ1 and we follow the same argument to get P + P
σ2 ∈ Q + E0(K), i.e.,
P + P σ2 lies above Γ1.
(ii) Since E/L3 has good reduction, see Lemma 6.2.4, it follows that E(L3) = E0(L3).
Moreover, since k = k, and char k 6= 3, the theory of formal groups implies that
E0(L3)/3E0(L3) = 0, see ([27], Chapter VII, Exercise 7.8). This means that the
multiplication-by-3 map is surjective on E(L3). Therefore, P = 3Q, for some Q ∈ E(L3),
61
and hence
∑2
i=0 P
σi3 = 3
∑2
i=0Q
σi3 ∈ 3E(K). Hence ∑2i=0 P σi3 ∈ E0(K) because
ΦK(E) ∼= Z/3Z. 2
Now assume that E/K has reduction type III∗. The special fiber of the minimal
proper regular model is as follows.
4
Ψ
Λ0 3
2
Θ0
Γ01
2 Θ2 3 Λ1
2
Θ1
Γ1 1
The multiplicity-1 identity component Γ0 is given by z = 0, the multiplicity-2 com-
ponent Θ0 is given by y
2
1 = 0, and the multiplicity-3 component Λ0 is given by x
3
1 = 0,
see ([28], Chapter IV, §9).
Proposition 6.2.6. Assume that E/K has reduction type III∗. Let Lm = K(tm), where
tm = t
1/m, m ∈ {2, 3, 4}. Let Gal(Lm/K) = 〈σm〉.
(i) Let P ∈ E(L2) lie above a multiplicity-2 component Θi, i ∈ {0, 1, 2}. Then P +
P σ2 ∈ E0(K) when i = 0, 1, and P + P σ2 6∈ E0(K) when i = 2.
(ii) Let P ∈ E(L3) be a point lying above Λi, i ∈ {0, 1}. Then we have
∑2
l=0 P
σl3 ∈
E0(K) when i = 0, and
∑2
l=0 P
σl3 6∈ E0(K) when i = 1.
(iii) Let P ∈ E(L4) be a point lying above Ψ. Then we have
∑3
l=0 P
σl4 ∈ E0(K).
Proof: Let Q ∈ E(K) be a point lying above Γ1. Let τ ′Q : Emin → Emin be the
extension of the translation-by-Q automorphism τQ.
(i) Let P ∈ E(L2). If P lies above Θ0 : y21 = 0, then P + P σ2 ∈ E0(K), see Lemma
6.2.1.
So let P ∈ E(L2) lie above Θ1. We have τ ′∗Q (Γ1) = Γ0, and τ ′∗Q (Γ1).τ ′∗Q (Θ1) = Γ1.Θ1,
so τ ′∗Q (Θ1) = Θ0. Therefore, τQ(P ) + τQ(P )
σ2 ∈ E0(K), i.e., P + P σ2 + 2Q ∈ E0(K). In
other words, P + P σ2 ∈ E0(K).
Now let P ∈ E(L2) lie above Θ2. By virtue of Lemma 6.2.4, the reduction type of
E/L2 is I
∗
0. We showed in the last paragraph of the previous subsection that E/L2 has
62
a minimal Weierstrass equation
y2 + a1xy + a3y = x
3 + a2x
2 + a4x+ a6, ai ∈ OL2 ,
where x3 + a2x
2 + a4x + a6 = (x− tα1)(x− tα2)(x− tα3), αi ∈ OL2 . Now σ2 fixes one
of the αi’s and swaps the other two because E[2] 6⊂ E(K), see ([28], p. 390), so we can
assume without loss of generality that α1 ∈ OK and α2, α3 ∈ OL2 \ OK , moreover we
know P1 = (tα1, 0) 6∈ E0(K).
Since the reduction map
{0, Pi = (tαi, 0)|i = 1, 2, 3} → ΦL2(E)
is surjective, it follows that P1 lies above a non-identity multiplicity-1 component Γ
′ of Ek,
where E is the minimal proper regular model of E/L2.Moreover, σ2 fixes the components
Γ(0,0) and Γ
′ of E and swaps the other two multiplicity-1 irreducible components Γ∗, Γσ2∗ .
Note that Γ(0,0) and Γ
′ lie above Γ0 and Γ1 respectively.
Since [L2 : K] divides the multiplicities of Θ2 and Ψ, it is a known fact that Γ∗ and
Γσ2∗ are lying above Θ2 under the following morphism
N˜ → N := Norm(Emin ×OK OL2)→ Emin,
where N is the normalisation of Emin×OK OL2 , and N˜ is the minimal desingularisation
of N , see for example ([20], §10.4) and ([22], pp. 10-11). The model E is obtained from
N˜ by contracting the exceptional divisors, see Definition 2.2.12.
Therefore, if P lies above Θ2 in E
min, then it lies on either Γ∗ or Γσ2∗ in E . Since
ΦL2(E)
∼= Z/2Z × Z/2Z, we have P + P σ2 lies above Γ′ in Ek, and hence it lies above
Γ1 in E
min
k .
(ii) If P ∈ E(L3) lies above the multiplicity-3 component Λ0 : x31 = 0, then
∑2
l=0 P
σl3
is a non-singular point. This follows from Proposition 6.2.5 (ii).
Let P ∈ E(L3) be a point lying above Λ1. Since τ ′∗Q (Θ1) = Θ0, see (i), and τ ′∗Q (Θ1).τ ′∗Q (Λ1) =
Θ1.Λ1, therefore τ
′∗
Q (Λ1) = Λ0. Thus
∑2
l=0 τQ(P )
σl3 ∈ E0(K). In other words,∑2l=0 P σl3+
3Q ∈ E0(K), which means ∑2l=0 P σl3 6∈ E0(K).
(iii) Lemma 6.2.4 shows that E/L4 has good reduction. It follows that E(L4) =
E0(L4). Moreover, since k = k, and char k 6= 2, the theory of formal groups implies
that E0(L4)/4E0(L4) = 0, see ([27], Chapter VII, Exercise 7.8). Therefore, P = 4Q, for
some Q ∈ E(L4) and hence
∑3
l=0 P
σl4 = 4
∑3
l=0Q
σl4 ∈ 4E(K), and we are done because
ΦK(E) ∼= Z/2Z. 2
Remark 6.2.7. When E/K has reduction type III∗, the above Proposition implies
that δ2(Θi) = 0¯ when i ∈ {0, 1}, and δ2(Θ2) = 1¯. Moreover, δ3(Λi) = i¯, i ∈ {0, 1}, and
δ4(Ψ) = 0¯.
63
Now assume that E/K has reduction type II∗. The special fiber of the minimal
proper regular model is as follows.
6
Λ1 3
2
Θ0
Γ0 1
Λ0 3
4
Ψ0
5 Ψ1 4
2
Θ1
Proposition 6.2.8. Assume that E/K has reduction type II∗. Let Lm = K(tm), where
tm = t
1/m. Let Gal(Lm/K) = 〈σm〉. Let P ∈ E(Lm) lie above a component of multi-
plicity m. Then
∑m−1
i=0 P
σim ∈ E0(K). In particular, δm(Γ) = 0¯, for every multiplicity-m
irreducible component Γ of Emink , m ∈ {1, 2, 3, 4}.
Proof: That is straightforward because
∑m−1
i=0 P
σim ∈ E(K), and E(K) = E0(K) for
reduction type II∗. 2
We conclude this chapter by stating the following direct corollary.
Corollary 6.2.9. Let E/K be an elliptic curve with minimal proper regular model Emin.
Let Lm, σm and δm, where m ∈ {1, 2, 3, 4}, be as above. Let Q ∈ E(K). Assume that
there exists a multiplicity-m irreducible component Θ in Emink such that δm(Θ) = ψE(Q).
Then there exists a point P ∈ E(Lm) lying above Θ such that Q =
∑m−1
i=0 P
σim .
Proof: The statement is clear for m = 1 by taking P = Q.
So assume m ≥ 2. Let P ′ ∈ E(Lm) be a point above Θ, see Proposition 5.2.1.
Then
∑m−1
i=0 P
′σim reduces to the same multiplicity-1 irreducible component as Q because
δm(Θ) = ψE(Q). Therefore, R := Q −
∑m−1
i=0 P
′σim ∈ E0(K). Since char k 6= 2, 3 and
k = k, we have E0(K) = mE0(K), see ([27], Chapter VII, Exercise 7.8). Thus R = mT
for some T ∈ E0(K).
Now consider the extension τ ′T : E
min → Emin of the translation-by-T automor-
phism τT . The multiplicity-1 irreducible components of E
min
k are fixed under τ
′
T , the
projection formula then implies that τ ′T fixes every irreducible component of E
min
k . Set
P := τT (P
′) = T + P ′. Then
∑m−1
i=0 P
σim = mT +
∑m−1
i=0 P
′σim = Q. 2
64
Chapter 7
Counting minimal degree-n-models
Let K be a Henselian discrete valuation field with ring of integers OK . We fix a uni-
formiser t. We denote the normalised valuation on K by ν. We assume that the residue
field k is algebraically closed and that char k 6= 2, 3.
Let C be a smooth genus one curve over K defined by a genus one equation φ of
degree n. We assume that C(K) 6= ∅. In this chapter we count the number of minimal
degree-n-models for C → Pn−1K up to isomorphism, where n ∈ {2, 3, 4}.
7.1 Counting results
Let E be the Jacobian of C. Since E ∼=K C, we can assume that φ is the defining
equation of a morphism E → Pn−1K determined by a divisor class [H], where H is a
hyperplane section divisor on E. The divisor H is linearly equivalent to (n− 1).0E + P
for some P ∈ E(K).
The homomorphism ψE is the surjective homomorphism in the following short exact
sequence
0→ E0(K)→ E(K) ψE−→ ΦE(K)→ 0.
We recall that ΦmK(E), m ≥ 1, is the set of multiplicity-m components in the special
fiber of the minimal proper regular model Emin of E. The main result of this chapter is
the following theorem.
Theorem 7.1.1. Let E/K be an elliptic curve and let P ∈ E(K). Let E → Pn−1K
be the morphism determined by the divisor class [(n − 1).0E + P ], n ∈ {2, 3, 4}. Let
δm : Φ
m
K(E) → ΦK(E) be the function defined in Chapter 6. Then there is a bijection
between the set of minimal degree-n-models for E → Pn−1K up to isomorphism and the
disjoint union of the following sets.
65
(i) The set of unordered n-tuples
S1(n) = {(a1, . . . , an) : ai ∈ ΦK(E) | a1 + . . .+ an = ψE(P )},
(ii) the set Sn = {a ∈ ΦnK(E) | δn(a) = ψE(P )},
and
if n ≥ 3
(iii) the set S(n−1,1) = {(a, b) ∈ Φn−1K (E)× ΦK(E) | δn−1(a) + b = ψE(P )},
and
if n = 4
(iv) the set S(2,1,1) = {(a, b, c) ∈ Φ2K(E)× ΦK(E)× ΦK(E) | δ2(a) + b+ c = ψE(P )},
where the order of b and c is immaterial,
(v) the set of unordered pairs
S(2,2) = {(a, b) ∈ Φ2K(E)× Φ2K(E) | δ2(a) + δ2(b) = ψE(P )}.
The proof will follow directly from Theorem 7.2.3 stated below.
The sets defined in the theorem above depend on the point P ∈ E(K). Now we get
the following corollaries as direct consequences of Theorem 7.1.1
Corollary 7.1.2. Let E/K be an elliptic curve and let P ∈ E(K). Let E → Pn−1K be
the morphism determined by the divisor class [(n − 1).0E + P ], n ∈ {2, 3, 4}. Assume
that E has one of the reduction types Im,m ≥ 0, II, III, or IV. Let m = #ΦK(E). The
number of minimal degree-n-models for E → Pn−1K is Nn where
(i) if n = 2
N2 =

(m+ 1)/2 if 2 - m
m/2 + 1 if 2 | m and ψE(P ) ∈ 2ΦK(E)
m/2 if 2 | m and ψE(P ) 6∈ 2ΦK(E)
(ii) if n = 3
N3 =

(m+ 1)(m+ 2)/6 if 3 - m
m(m+ 3)/6 + 1 if 3 | m and ψE(P ) ∈ 3ΦK(E)
m(m+ 3)/6 if 3 | m and ψE(P ) 6∈ 3ΦK(E)
66
(iii) if n = 4
N4 =

(m+ 1)(m+ 2)(m+ 3)/24 if 2 - m
3 if m = 2 and ψE(P ) = 0¯
m(m+ 2)(m+ 4)/24 + 2 if 2 | m, m 6= 2 and ψE(P ) ∈ 4ΦK(E)
m(m+ 2)(m+ 4)/24 + 1 if 2 | m and ψE(P ) ∈ 2ΦK(E) \ 4ΦK(E)
m(m+ 2)(m+ 4)/24 if 2 | m and ψE(P ) 6∈ 2ΦK(E).
Proof: Since for the given reduction types we have ΦmK(E) = ∅ when m ≥ 2, the sets
Sn, S(n−1,1), S(2,1,1) and S(2,2) of Theorem 7.1.1 are empty. In other words, Nn = #S1(n).
The numbers stated above are the possible cardinalities of S1(n). 2
For a ∈ Φ2K(E) we set
S(2,1,1)(a) = {(b, c) : b, c ∈ ΦK(E) | b+ c = ψE(P )− δ2(a)}.
Lemma 7.1.3. The following equalities hold
(i) #S(n−1,1) = #Φn−1K (E) when n ≥ 3.
(ii) #S(2,1,1) =
∑
a∈Φ2K(E) S(2,1,1)(a).
Proof: (i) We consider the projection map S(n−1,1) → Φn−1K (E) which sends an element
(a, b) ∈ Φn−1K (E)×ΦK(E) to a. It is surjective and we want to show the injectivity. Let
(a1, b1), (a2, b2) be two elements in S(n−1,1) with the same image, i.e., a1 = a2. From the
definition of S(n−1,1) we have δn−1(a1) + b1 = δn−1(a2) + b2 = ψE(P ). Therefore, b1 = b2,
and the two sets are in bijection.
(ii) Direct calculation by computing the number of triples in which each a ∈ Φ2K(E)
lies. 2
Corollary 7.1.4. Let E/K be an elliptic curve and let P ∈ E(K). Assume that E
has reduction type T ∈ {I∗m, m ≥ 0, IV∗, III∗, II∗}. Let E → Pn−1K be the morphism
determined by the divisor class [(n − 1).0E + P ], n ∈ {2, 3, 4}. Let Nn denote the
number of minimal degree-n-models for E → Pn−1K . Then Nn is determined according to
the following tables
(i) if E has reduction type I∗2m,m ≥ 0, then
67
ψE(P ) (0¯, 0¯) (1¯, 1¯) (0¯, 1¯) (1¯, 0¯)
N2 m+ 5 m+ 2 2 2
N3 2m+ 6
N4 (m+ 4)
2 (m+ 2)(m+ 5) 4m+ 10
(ii) if E has reduction type I∗2m+1,m ≥ 0, then
ψE(P ) 0¯ 2¯ 1¯ 3¯
N2 m+ 4 2
N3 2m+ 7
N4 (m+ 3)(m+ 6) (m+ 4)
2 4m+ 12
(iii) if E has reduction type IV∗, then
ψE(P ) 0¯ 1¯ 2¯
N2 3
N3 8 6 6
N4 14
(iv) if E has reduction type III∗, then
ψE(P ) 0¯ 1¯
N2 4 2
N3 6 6
N4 15 10
(v) if E has reduction type II∗, then
N2 3
N3 5
N4 10
68
Proof: We recall that the number of minimal degree-2-models for E → P1K is the
cardinality of the disjoint union of the sets S1(2) and S2. The number of minimal degree-
3-models for E → P2K is the cardinality of the disjoint union of S1(3), S(2,1), and S3. The
number of minimal degree-4-models for E → P3K is the cardinality of the disjoint union
of S1(4), S(3,1), S(2,2), S(2,1,1), and S4. So we have to find the cardinality of each set when
E has one of the reduction types I∗m, IV
∗, III∗, or II∗.
(i) When E has reduction type I∗2m,m ≥ 0, the components group ΦK(E) is iso-
morphic to Z/2Z × Z/2Z. We have #S1(2) = 4 when ψE(P ) = (0¯, 0¯) and #S1(2) = 2
otherwise. We have #S1(3) = 5, moreover #S1(4) = 11 when ψE(P ) = (0¯, 0¯), and
#S1(4) = 8 otherwise. Moreover, #S(2,1) = #Φ
2
K(E) = 2m + 1, see Lemma 7.1.3. The
sets Si, i = 3, 4, and S(3,1) are empty.
To compute the size of S2, S(2,1,1), and S(2,2) we need to recall some results from
Chapter 6. The multiplicity-2 components of Emink consists of a sequence V−1 : {y21 =
0}, V0, V1, . . . , V2m−1, where δ2(Vi) = (0¯, 0¯) if i is odd and it is equal to (1¯, 1¯) otherwise,
see Corollary 6.2.3. Therefore, S2 = ∅ if ψE(P ) = (0¯, 1¯) or (1¯, 0¯), and #S2 = m + 1 if
ψE(P ) = (0¯, 0¯), and #S2 = m if ψE(P ) = (1¯, 1¯).
Now if ψE(P ) = δ2(Vi), then #S(2,1,1)(Vi) = 4, otherwise #S(2,1,1)(Vi) = 2, see
Lemma 7.1.3. Therefore, if ψE(P ) = (0¯, 1¯) or (0¯, 1¯), then #S(2,1,1)(Vi) = 2 for each i and
#S(2,1,1) = 2(2m + 1), see Lemma 7.1.3. If ψE(P ) = (0¯, 0¯)(= (1¯, 1¯) respectively), then
(#S(2,1,1)(V2i−1),#S(2,1,1)(V2i)) = (4, 2), ((2, 4) respectively), i = 0, . . . ,m. Therefore, we
have #S(2,1,1) = 4(m+ 1) + 2m, (2(m+ 1) + 4m respectively), see Lemma 7.1.3.
Again if ψE(P ) = (0¯, 1¯) or (1¯, 0¯), then S(2,2) = ∅. If ψE(P ) = (0¯, 0¯), then #S(2,2) =∑m+1
i=1 i+
∑m
i=1 i = (m+ 1)
2. If ψE(P ) = (1¯, 1¯), then #S(2,2) = m(m+ 1).
(ii) When E has reduction type I∗2m+1,m ≥ 0, the components group ΦK(E) = Z/4Z.
We have #S1(3) = 5. Lemma 7.1.3 implies that #S(2,1) = #Φ
2
K(E) = 2m+2. Moreover,
Si, i = 3, 4, and S3,1 are empty sets.
Now S1(2) = 3 when ψE(P ) = 0¯ or 2¯, and #S1(2) = 2 otherwise. We have #S1(4) =
10 when ψE(P ) = 0¯, #S1(4) = 9 when ψE(P ) = 2¯, and #S1(4) = 8 otherwise.
We recall that the multiplicity-2 components of Emink consists of a sequence V−1 :
{y21 = 0}, V0, V1, . . . , V2m, where δ2(Vi) = 0¯ if i is odd and it is equal to 2¯ otherwise,
see Corollary 6.2.3. Therefore, if ψE(P ) = 1¯ or 3¯, then S2 and S(2,2) are empty sets. If
ψE(P ) = 0¯ or 2¯, then #S2 = m+ 1.
If ψE(P ) − δ2(Vi) ∈ 2ΦK(E), then #S(2,1,1)(Vi) = 3. Otherwise, #S(2,1,1)(Vi) = 2.
Therefore, if ψE(P ) = 1¯ or 3¯, then #S(2,1,1)(Vi) = 2 for each i, hence #S(2,1,1) =
2(2m + 2), see Lemma 7.1.3. If ψE(P ) = 0¯ or 2¯, then #S(2,1,1)(Vi) = 3 for each i.
Therefore, #S(2,1,1) = 3(2m+ 2).
If ψE(P ) = 1¯ or 3¯, then S(2,2) = ∅. If ψE(P ) = 0¯, then #S(2,2) = (m + 1)(m + 2). If
69
ψE(P ) = 2¯, then #S(2,2) = (m+ 1)
2.
(iii) When E has reduction type IV∗, we have ΦK(E) = Z/3Z. Therefore, #S1(2) = 2
and #S1(4) = 5, moreover #S1(3) = 4 when ψE(P ) = 0¯ and #S1(3) = 3 otherwise.
We recall that #S3 = 1 if ψE(P ) = 0¯, and S3 = ∅ otherwise, see Proposition
6.2.5 (ii). Moreover, Φ4K(E) = ∅ and Φ2K(E) = {Θi, i = 0, 1, 2}, where δ2(Θ0) = 0¯,
δ2(Θ1) = 2¯, and δ2(Θ2) = 1¯, see Proposition 6.2.5. Therefore, Lemma 7.1.3 implies that
#S(2,1) = #Φ
2
K(E) = 3, and #S(3,1) = #Φ
3
K(E) = 1. We have S4 = ∅, #S2 = 1, and
#S(2,2) = 2. Furthermore, S(2,1,1)(a) = 2 for every a ∈ Φ2K(E), therefore #S(2,1,1) = 6.
(iv) When E has reduction type III∗, we have ΦK(E) = Z/2Z. Therefore, #S1(2) = 2
if ψE(P ) = 0¯, and #S1(2) = 1 otherwise. #S1(3) = 2, moreover we have #S1(4) = 3
when ψE(P ) = 0¯, and #S1(4) = 2 otherwise.
Recall that Φ2K(E) = {Θi, i = 0, 1, 2}, where δ2(Θi) = 0¯ when i = 0, 1, and δ2(Θ2) =
1¯. Moreover, Φ3K(E) = {Λ1,Λ2}, where δ3(Λi) = i¯, and Φ4K(E) = {Ψ}, where δ4(Ψ) = 0¯,
see Proposition 6.2.6. By virtue of Lemma 7.1.3, we have #S(2,1) = #Φ
2
K(E) = 3, and
#S(3,1) = #Φ
3
K(E) = 2. If ψE(P ) = 0¯, then #S2 = 2, #S2,2 = 4, #S(2,1,1) = 5, and
#S4 = 1. Otherwise, we have #S2 = 1,#S2,2 = 2, #S(2,1,1) = 4, and S4 = ∅. We always
have #S3 = 1.
(v) When E has reduction type II∗, we have ΦK(E) = (0). Therefore, #S1(n) =
1, n ∈ {2, 3, 4}, see Corollary 7.1.2.
We have that #S(n−1,1) = #Φn−1K (E) = 2, n = 3, 4, see Lemma 7.1.3. Moreover,
#Sn = 2, n = 2, 3, 4. Finally, S(2,2) = 3 and S(2,1,1) = 2, see Proposition 6.2.8. 2
7.2 Constructing divisors and models
The aim of this section is to prove Theorem 7.1.1. Again we start with a smooth genus
one curve C/K where n ∈ {2, 3, 4}. We assume that the equation of C is obtained as
an equation for the morphism E → Pn−1K determined by [(n − 1).0E + P ], where E is
the Jacobian of C and P ∈ E(K). We set
S2 = S1(2) ∪ S2, S3 = S1(3) ∪ S3 ∪ S(2,1) and S4 = S1(4) ∪ S4 ∪ S(3,1) ∪ S(2,1,1) ∪ S(2,2).
Then we define the map
λn : {minimal degree-n-models for C up to isomorphism} → Sn; (C, α) 7→ (Γ1, . . . ,Γm),
where
(i) (Γ1, . . . ,Γm) consists of the irreducible components of the strict transform of Ck in
Emink .
70
(ii) If m ≥ 2, then multk(Γ1) ≥ . . . ≥ multk(Γm).
(iii) If Γ is the strict transform of a component Γ′ in Ck, then Γ appears in (Γ1, . . . ,Γm)
as many times as degk Γ
′. In particular, we have
∑m
i=1multk(Γi) = n.
We will prove that the map λn is a bijection and hence Theorem 7.1.1 follows. Recall
that the type of an irreducible component Γ of the special fiber of a degree-n-model is
the pair (multk Γ, degk Γ).
Lemma 7.2.1. The map λn described above is well-defined.
Proof: We need to show that (i) if C and C ′ are minimal isomorphic degree-n-models for
C, then λn(C) = λn(C ′), and (ii) λ(C) ∈ Sn. To prove (i) we know that the special fibers
of any two isomorphic degree-n-models for C have the same irreducible components
with the same types, therefore the corresponding tuples of both models are the same,
see Theorem 5.1.4.
Now we want to prove (ii), in other words we want to show that if (Γ1, . . . ,Γm) is
the tuple associated to C, then ∑mi=1 δmi(Γi) = ψE(P ) where mi = multk(Γi). Let D
be a divisor on Emin such that C is obtained from Emin by contraction using D, see
Theorem 4.0.1. The divisor D intersects the Γi’s and no other components. We have
D.Γj = mj degk Γ
′
j, where Γ
′
j is an irreducible component of Ck whose strict transform
in Emink is Γj, in particular if degk Γ
′
j = dj, then D intersects Γj in dj points which
might not be distinct. Let (x1, . . . , xm) be a tuple of all intersection points of D with
the irreducible components (Γ1, . . . ,Γm), where xi ∈ Γi. Note that xi may be repeated
in (x1, . . . , xm) if Γi is the strict transform of a component whose degree is greater than
1. Moreover, we have D|E ∼ (n− 1).0 + P.
Let mr = max{mj := multk Γj : 1 ≤ j ≤ m}. It is clear that for Γj, j = 1, . . . ,m,
the multiplicity mj ∈ {1,mr}, since otherwise we will have 1 < mj < mr which implies
that mr ≥ 3, hence mr + mj ≥ 5 which contradicts that
∑m
i=1mi = n ≤ 4. Let
Lj = K(t
1/mj) and Gal(Lj/K) = 〈σj〉. By virtue of Proposition 5.2.1, there exists a
closed point Pj ∈ E(Lj) such that [K(Pj) : K] = mj, and {Pj} ∩ Γj = {xj}. Let E be
the minimal proper regular model of E/Lr. Since xj lies on one and only one component
of Emink , then there are exactly mj points {y1, . . . , ymj} of E lying above xj, and each
of these points lies on a multiplicity-1 component of Ek, see ([20], Remark 10.4.8). Now
we view D as a divisor on E . We have
P˜ = (sumD|E)∼ = sumD|Ek =
m∑
j=1
mj∑
l=1
yl =
m∑
j=1
mj∑
l=1
P˜
σlj
j .
71
The second equality holds because D intersects Ek only in multiplicity-1 components.
Therefore, we have P −∑mj=1∑mjl=1 P σljj ∈ E0(K). Applying the surjective group homo-
morphism ψE : E(K) → ΦK(E), we get ψE(P ) =
∑m
j=1 ψE(
∑mj
l=1 P
σlj
j ) =
∑m
j=1 δmj(Γj).
2
Lemma 7.2.2. Let (Γ1, . . . ,Γm) ∈ Sn. Then there exists a divisor D on Emin such that:
(i) D|E is K-rational.
(ii) D.Γi = di.multk Γi, where di is the number of times Γi appears in (Γ1, . . . ,Γm).
(iii) D|E ∼ (n− 1).0 + P.
Proof: Assume that (Γ1, . . . ,Γn) ∈ S1(n). Let xi ∈ Γi, i = 1, . . . , n − 1, be a point
defined over k. Hensel’s Lemma allows us to lift xi to a point Pi ∈ E(K), i = 1, . . . , n−1.
Set Pn = P −
∑n−1
i=1 Pi ∈ E(K). Note that Pn lies above Γn because ψE(Pn) = ψE(P )−∑n−1
j=1 ψE(Pi) = ψE(P )−
∑n−1
i=1 δ1(Γi) = δ1(Γn), where the last equality follows from the
definition of S1(n). Set the divisor D to be
∑n
i=1 {Pi} on Emin, where {Pi} is the Zariski
closure of Pi in E
min.
Now let Km = K(t
1/m) and Gal(Km/K) = 〈σm〉.
Assume that (Γn) ∈ Sn. Since δn(Γn) = ψE(P ), Corollary 6.2.9 shows that there
exists a point Q ∈ E with [K(Q) : K] = n, Q˜ ∈ Γn and
∑n
i=1Q
σin = P. Set D = {Q}.
Assume that (Γn−1,Γ) ∈ S(n−1,1), n ≥ 3. Let Q′ be a point which reduces on Γ. We
have δn−1(Γn−1) = ψE(P )− δ1(Γ) = ψE(P −Q′). Again Corollary 6.2.9 shows that there
exists a point Q ∈ E with [K(Q) : K] = n − 1, Q˜ ∈ Γn−1, and
∑n−1
i=1 Q
σin−1 = P − Q′.
Set D = {Q′}+ {Q}.
Assume that (Θ,Γ1,Γ2) ∈ S(2,1,1). Let Pi ∈ E(K) be such that P˜i ∈ Γi. We have
δ2(Θ) = ψE(P − P1 − P2). Again there exists a point Q ∈ E with [K(Q) : K] = 2,
Q˜ ∈ Θ, and Q+Qσ2 = P − P1 − P2. Set D = {P1}+ {P2}+ {Q}.
Assume that (Θ1,Θ2) ∈ S(2,2). Let P ′ ∈ E(K) be such that δ2(Θ1) = ψE(P ′).
According to Corollary 6.2.9, there exists Qi ∈ E with [K(Qi) : K] = 2, Q˜i ∈ Θi,
Q1 +Q
σ2
1 = P
′, and Q2 +Q
σ2
2 = P − P ′ . Set D = {Q1}+ {Q2}. 2
Theorem 7.2.3. The map λn is a bijection.
Proof: We proved that λn is well defined in Lemma 7.2.1. To prove that λn is injective,
let (C1, α1) and (C2, α2) be two minimal degree-n-models for C. Assume that λn(C1) =
λn(C2). We want to show that (C1, α1) and (C2, α2) are isomorphic. The fact that they
have the same corresponding tuples implies that for an irreducible component Γ of (C2)k,
72
we have (α−12 α1)
∗Γ is an irreducible component of (C1)k with the same type, therefore
(C1, α1) and (C2, α2) are isomorphic, see Theorem 5.1.4.
Now we will prove that λn is surjective. So assume that (Γ1, . . . ,Γm) ∈ Sn and we
want to construct a minimal degree-n-model (C, α) whose image under λn is (Γ1, . . . ,Γm).
If Γi appears di times in (Γ1, . . . ,Γm), then Ck should contain an irreducible component
Γ′i, whose strict transform in E
min
k is Γi, such that degk Γ
′
i = di. By virtue of Lemma
7.2.2, there exists a divisor D on Emin such that D|E is K-rational, D.Γi = di.multk Γi,
and D|E ∼ (n− 1).0 + P. Consider the following S-model for C
C := Proj(
∞⊕
m=0
H0(Emin,OEmin(mD))).
The model C is given by a genus one equation of degree n, moreover it is minimal
because it is obtained by contracting components in Emin, see Theorem 4.2.3. The
special fiber Ck consists of the components in (Γ1, . . . ,Γm) because D intersects these
components in Emink and intersects no other components. Moreover, each component of
Ck has degree equal to the number of iterations of its strict transform in (Γ1, . . . ,Γm)
because D.Γi = di.multk Γi. We obtain α from the linear equivalence of D|E and
(n− 1).0E + P. 2
73
Chapter 8
Counting models over Q
Let p ≥ 5 be a prime number, and n ∈ {2, 3, 4}. Let Qunp be the maximal unramified
extension of the p-adic field Qp. Let Zp be the ring of p-adic integers and Zunp the ring
of integers of Qunp .
In this chapter we will be interested in attacking the global question. More precisely,
let C be a smooth genus one curve over Q defined by an integral genus one equation
of degree n, we define a minimal global degree-n-model (C, α) for C → Pn−1Q to be a
degree-n-model C → SpecZ for C → Pn−1Q such that C → SpecZp is minimal for every
prime p, see §3.1.
It is known that a minimal global degree-1-model for C/Q, i.e., a Weierstrass model,
exists and is unique up to isomorphism. A proof of the existence of a minimal global
degree-n-model for C/Q when n ∈ {2, 3} can be found in ([14], Theorem 2.6). An algo-
rithm which proves the existence of minimal global degree-n-models when n ∈ {1, 2, 3, 4}
can be found in [11].
The problem stated above can be tackled locally by considering C as a curve over Qp
and looking at minimal degree-n-models for C → Pn−1Qp up to SpecZp-isomorphism at
each prime p. It turns out that we need only to investigate a finite set of primes. We need
to overcome two obstacles: (i) the residue fields Fp are not algebraically closed which
means that we can not use the results we obtained in the previous chapters directly (ii)
collecting the local data we obtain at each prime to count minimal global models up to
SpecZ-isomorphism.
We will show that the split reduction types are no different from the geometric
case, i.e., when the residue field is algebraically closed. Then we move to the non-split
reduction types and use the fact that after a finite unramified base change the reduction
becomes split. In fact, we make use of the action of the Galois group Gal(Fp/Fp) on the
irreducible components of the minimal proper regular model.
74
The last remark to make is that considering C as a curve over Qunp rather than over
Qp, the number of degree-n-models for C → Pn−1Qunp up to SpecZunp -isomorphism is deter-
mined according to Theorem 7.1.1 because the residue field Fp is algebraically closed.
Thus we only need to figure out how many of these models are SpecZunp -isomorphic to
ones defined over Zp.
8.1 Fp-rational divisors
We write Gp for Gal(Fp/Fp). Let C be a smooth genus one curve over Qp, such that
C(Qp) 6= ∅, with Jacobian elliptic curve E and minimal proper regular model Emin.
Let C → SpecZunp be a minimal degree-n-model for C → Pn−1Qp . Set IrrC to be the tuple
consisting of the irreducible components of the strict transform of CFp in EminFp such
that if Γ is an irreducible component of CFp , then the strict transform of Γ appears in
IrrC as many times as degFp Γ. It is understood that only multiplicity-m components,
where m = 1, 2, can have degrees greater than 1 in CFp . The order of the tuple IrrC
is immaterial. The group Gp acts on IrrC in the obvious way, namely (Γ1, . . . ,Γl)σ =
(Γσ1 , . . . ,Γ
σ
l ) for σ ∈ Gp.
We recall that C is obtained from Emin → SpecZunp by contraction via a divisor D,
see Theorem 4.0.1. The divisor D intersects each irreducible component of IrrC and
meets no other components in EminFp , i.e., for an irreducible component Γ of E
min
Fp , we
have D ∩ Γ 6= ∅ if and only if Γ ∈ IrrC. The defining genus one equation φ of C is
obtained as a dependence relation in a finitely generated free Zunp -module of the form
H0(Emin,OEmin(mD)), for some m ≥ 1, see Theorem 4.2.3.
In this section we give a necessary and sufficient condition for C to be SpecZunp -
isomorphic to a degree-n-model which is defined over Zp, in other words the condition
under which it is possible to construct a divisor D′ on Emin → SpecZunp such that
OEmin(D) ∼= OEmin(D′) and D′|C is Qp-rational. In fact, this condition will turn out to
be that IrrC is Gp-invariant. The main result for counting degree-n-models over Zp is
the following theorem.
Theorem 8.1.1. Let C be a smooth curve over Qp defined by an integral genus one
equation of degree n with C(Qp) 6= ∅. Let Emin → SpecZunp be the minimal proper
regular model of the Jacobian elliptic curve E. Let (C → SpecZunp , α) be a minimal
degree-n-model for C → Pn−1Qp . Then (C, α) is SpecZunp -isomorphic to a minimal degree-
n-model defined over Zp if and only if IrrC is Gp-invariant.
The next section is devoted to the proof of Theorem 8.1.1. Now we get some ap-
plications of Theorem 8.1.1. We will count minimal degree-n-models for C which are
75
defined over Zp. That is the number of all the combinatorial possibilities for IrrC to be
Gp-invariant. Recall that the genus one equation defining C can be considered as an equa-
tion for a morphism E → Pn−1Qp determined by a divisor class of the form [(n−1).0E+P ]
for some P ∈ E(Qp). This point P reduces to a point on a multiplicity-1 irreducible
component in EminFp defined over Fp.
Corollary 8.1.2. Let E/Qp be an elliptic curve and let P ∈ E(Qp). Assume that E
has split reduction. Let E → Pn−1Qp be the morphism determined by the divisor class
[(n − 1).0E + P ], n ∈ {2, 3, 4}. Let δm : ΦmQunp (E) → ΦQunp (E) be the function defined
in Chapter 6. Then there is a bijection between the set of minimal degree-n-models for
E → Pn−1Qp up to isomorphism and the union of the sets given in Theorem 7.1.1.
Proof: If we consider E as a curve over Qunp , then the number of minimal degree-n-
models up to SpecZunp -isomorphism is the cardinality of the disjoint union of the sets
stated in Theorem 7.1.1. But Theorem 8.1.1 shows that each of these models is isomor-
phic to a minimal degree-n-model with coefficients in Zp because all the components of
EminFp are defined over Fp. 2
Remark 8.1.3. If E has one of the reduction types
Im,m ∈ {0, 1, 2}, II, III, III∗, II∗,
then the reduction is always split, in particular all irreducible components of EminFp are
defined over Fp. Therefore, the number of minimal degree-n-models for these reduction
types is determined according to Corollary 8.1.2.
For the rest of this section we will be concerned with non-split reduction types.
We will count the set of minimal degree-n-models C → SpecZunp for which IrrC is Gp-
invariant. We will denote the number of multiplicity-1 irreducible components of EminFp
which are defined over Fp by cp, this is the Tamagawa number of the Jacobian E/Qp.
Corollary 8.1.4. Let E/Qp be an elliptic curve and let P ∈ E(Qp). Let E → Pn−1Qp
be the morphism determined by the divisor class [(n − 1).0E + P ]. Assume that E has
non-split reduction. Then the number of minimal degree-n-models for E → Pn−1Qp up to
isomorphism is determined according to the following table.
76
cp n = 2 n = 3 n = 4
I2m+1 1 m+ 1 m+ 1 (m+ 1)(m+ 2)/2
I2m, P ∈ E0(Qp) 2 m+ 1 m+ 1 (m+ 1)(m+ 2)/2
I2m, P 6∈ E0(Qp) 2 1 m+ 1 m+ 1
IV 1 2 2 3
I∗0 1 2 3 4
I∗2m+1, m ≥ 0, P ∈ E0(Qp) 2 m+ 4 2m+ 5 (m+ 3)(m+ 4)
I∗2m+1, m ≥ 0, P 6∈ E0(Qp) 2 m+ 2 2m+ 5 (m+ 2)(m+ 4)
I∗2m, m ≥ 0, P ∈ E0(Qp) 2 m+ 3 2m+ 4 (m+ 2)(m+ 4)
I∗2m, m ≥ 0, P 6∈ E0(Qp) 2 m+ 2 2m+ 4 (m+ 2)(m+ 3)
IV∗ 1 3 4 8
Recall that ψE is the surjective group homomorphism in the short exact sequence
0→ E0(Qp)→ E(Qp) ψE−→ ΦQp(E)(Fp)→ 0,
and that we fix an isomorphism ΦQunp (E)(Fp) ∼= Z/2Z × Z/2Z when E has reduction
I∗2m, and ΦQunp (E)(Fp) ∼= Z/lZ for other reduction types.
Proof of Corollary 8.1.4:
Multiplicative reduction: It is known that if E/Qp has non-split multiplicative
reduction Im, m ≥ 3, over Qp, then there exists an unramified quadratic extensionK/Qp
such that E/K has split multiplicative reduction Im.
For reduction type I2m+1 we have cp = 1. Therefore, the identity component Γ0
is the only multiplicity-1 component defined over Fp. We have ψE(P ) = 0¯. The two
intersection points of Γ0 with the other irreducible components of E
min
Fp are switched
under the action of some σ ∈ Gp. It follows that σ carries the ith component of EminFp to
the (2m+ 1− i)th component.
For reduction type I2m we have cp = 2. One of the two irreducible components
defined over Fp is the identity component. Again Gp carries the ith component to the
(2m− i)th component, therefore the other irreducible component defined over Fp is the
mth component. We have ψE(P ) is either 0¯ or m.
Now the tuple IrrC corresponding to a minimal degree-n-model C for C is Gp-invariant
if and only if it consists of components of the form determined by the following table.
77
n = 2 n = 3 n = 4
Im, ψE(P ) = 0¯ (¯i,m− i¯) (0¯, i¯, m− i¯) (¯i,m− i¯, j¯, m− j¯)
Im, m : even, ψE(P ) = m/2 (0¯,m/2) (m/2, i¯, m− i¯). (0¯,m/2, i¯, m− i¯)
The numbers given in the statement of the corollary are the cardinalities of the sets
consisting of the tuples given in the table above.
Additive reduction: Now we consider non-split additive reduction types. Let C
be a minimal degree-n-model for E → Pn−1Qp determined by [(n − 1).0E + P ]. Let Nn
be the number of minimal degree-n-models which are defined over Zp, up to SpecZp-
isomorphism, i.e., Nn is the number of minimal degree-n-models, up to SpecZp-isomorphism,
whose corresponding tuples are Gp-invariant.
Reduction IV: If E has non-split reduction of type IV, then cp = 1. We have
ψE(P ) = 0¯. The group Gp switches the two non-identity components. When n = 2, the
tuple IrrC is Gp-invariant if and only if it is either (0¯, 0¯) or (1¯, 2¯). Therefore, N2 = 2.
When n = 3, IrrC is Gp-invariant if and only if it is either (0¯, 0¯, 0¯) or (0¯, 1¯, 2¯). Hence
N3 = 2. When n = 4, IrrC is Gp-invariant if and only if IrrC ∈ {(0¯, 0¯, 0¯, 0¯), (0¯, 0¯, 1¯, 2¯),
(1¯, 2¯, 1¯, 2¯)}. Hence N4 = 3.
Reduction I∗0: If E has reduction type I
∗
0, then
cp = 1 +#{α ∈ Fp : f(α) = 0}
where f is a polynomial of degree 3 in the coefficients of the defining polynomial of E,
see ([28], p. 367). Moreover, f has distinct roots in Fp. Thus if the reduction is non-split,
then either cp = 1 or cp = 2. Moreover, the multiplicity-2 component Θ is defined over
Fp, i.e., it is isomorphic to P1Fp . In addition, δ2(Θ) = (0¯, 0¯), see Corollary 6.2.3.
When cp = 1, we have ψE(P ) = (0¯, 0¯) and Gp swaps the 3 non-identity components.
When cp = 2, we fix an isomorphism ΦQunp (E)(Fp) ∼= Z/2Z× Z/2Z such that (1¯, 1¯) cor-
responds to the non-identity component which is defined over Fp. Then we have ψE(P )
is either (0¯, 0¯) or (1¯, 1¯), and Gp switches the two other components. The multiplicity-2
component Θ is always fixed under the action of Gp.
Assume that cp = 1. For the case n = 2, IrrC is Gp-invariant if and only if it
is either ((0¯, 0¯), (0¯, 0¯)) or the multiplicity-2 component Θ. When n = 3, IrrC is Gp-
invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯)), ((0¯, 1¯), (1¯, 0¯), (1¯, 1¯)) or ((0¯, 0¯),Θ).
When n = 4, IrrC is Gp-invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯), (0¯, 0¯)),
((0¯, 0¯), (0¯, 1¯), (1¯, 0¯), (1¯, 1¯)), ((0¯, 0¯), (0¯, 0¯),Θ) or (Θ,Θ). Hence N2 = 2, N3 = 3 and
N4 = 4.
Now assume that cp = 2. Assume moreover that ψE(P ) = (0¯, 0¯). When n =
2, IrrC is Gp-invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯)), ((1¯, 1¯), (1¯, 1¯)) or the
78
multiplicity-2 component Θ. When n = 3, IrrC is Gp-invariant if and only if it is either
((0¯, 0¯), (0¯, 0¯), (0¯, 0¯)), ((0¯, 0¯), (1¯, 1¯), (1¯, 1¯)), ((0¯, 1¯), (1¯, 0¯), (1¯, 1¯)) or ((0¯, 0¯),Θ).When n = 4,
IrrC is Gp-invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯), (0¯, 0¯)), ((0¯, 0¯), (0¯, 0¯), (1¯, 1¯), (1¯, 1¯)),
((0¯, 0¯), (0¯, 1¯), (1¯, 0¯), (1¯, 1¯)), ((0¯, 1¯), (1¯, 0¯), (0¯, 1¯), (1¯, 0¯)), ((1¯, 1¯), (1¯, 1¯), (1¯, 1¯), (1¯, 1¯)), ((1¯, 1¯), (1¯, 1¯),Θ),
((0¯, 0¯), (0¯, 0¯),Θ) or (Θ,Θ). Hence N2 = 3, N3 = 4 and N4 = 8.
Assume ψE(P ) = (1¯, 1¯). When n = 2, IrrC is Gp-invariant if and only if it is ei-
ther ((0¯, 0¯), (1¯, 1¯)) or ((0¯, 1¯), (1¯, 0¯)). When n = 3, IrrC is Gp-invariant if and only if
it is either ((0¯, 0¯), (0¯, 0¯), (1¯, 1¯)), ((0¯, 0¯), (0¯, 1¯), (1¯, 0¯)), ((1¯, 1¯), (1¯, 1¯), (1¯, 1¯)) or ((1¯, 1¯),Θ).
When n = 4, IrrC is Gp-invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯), (1¯, 1¯)),
((0¯, 0¯), (0¯, 0¯), (0¯, 1¯), (1¯, 0¯)), ((1¯, 1¯), (1¯, 1¯), (0¯, 1¯), (1¯, 0¯)), ((0¯, 0¯), (1¯, 1¯), (1¯, 1¯), (1¯, 1¯)), ((0¯, 0¯), (1¯, 1¯),Θ)
or ((0¯, 1¯), (1¯, 0¯),Θ). Hence N2 = 2, N3 = 4 and N4 = 6.
Reduction I∗m, m ≥ 1: If E has non-split reduction of type I∗m, m ≥ 1, then cp = 2.
The multiplicity-1 non-identity component Γ which is defined over Fp is the one attached
to the same multiplicity-2 component to which the identity component is attached. From
the description of the irreducible component Γ, it follows that it corresponds to an
element of order 2 in the components group, see Proposition 6.2.2, so it corresponds to
2¯ when m is odd. When m is even, we fix an isomorphism ΦQunp (E)(Fp) ∼= Z/2Z×Z/2Z
such that Γ corresponds to (1¯, 1¯). The group Gp switches the two other multiplicity-1
irreducible components. All the multiplicity-2 components are defined over Fp, i.e., each
of them is isomorphic to P1Fp , and hence they are fixed under the action of Gp.
(i) Reduction I∗2m+1: We start with non-split reduction of type I
∗
2m+1, m ≥ 0.
We recall that the multiplicity-2 components of EminFp are V−1, V0, V1, . . . , V2m where
δ2(Vi) = 0¯ if i is odd and it is equal to 2¯ otherwise, see Corollary 6.2.3.
Assume that ψE(P ) = 0¯. When n = 2, IrrC is Gp-invariant if and only if it is either
(0¯, 0¯), (2¯, 2¯), (1¯, 3¯), or Vi where i is odd. When n = 3, IrrC is Gp-invariant if and only
if it is either (0¯, 0¯, 0¯), (0¯, 2¯, 2¯), (0¯, 1¯, 3¯), (0¯, Vi) where i is odd, or (2¯, Vi) where i is even.
When n = 4, IrrC is Gp-invariant if and only if it is either (0¯, 0¯, 0¯, 0¯), (0¯, 0¯, 2¯, 2¯), (0¯, 0¯, 1¯, 3¯),
(2¯, 2¯, 1¯, 3¯), (2¯, 2¯, 2¯, 2¯), (1¯, 3¯, 1¯, 3¯) or if IrrC lies in
S(2,1,1)(Vi) =
 {(Vi, 0¯, 0¯), (Vi, 2¯, 2¯), (Vi, 1¯, 3¯)} if i is odd{(Vi, 0¯, 2¯)} if i is even
or if IrrC is (Vi, Vj) where i + j is even. Therefore, N2 = m + 4, N3 = 2m + 5 and
N4 = 6 + 4(m+ 1) + (m+ 1)(m+ 2) = (m+ 3)(m+ 4).
Now assume that ψE(P ) = 2¯. When n = 2, IrrC is Gp-invariant if and only if it is
either (0¯, 2¯) or Vi where i is even. When n = 3, IrrC is Gp-invariant if and only if it is
either (0¯, 0¯, 2¯), (2¯, 2¯, 2¯), (2¯, 1¯, 3¯), or if it is of the form (0¯, Vi) where i is even, or (2¯, Vi)
79
where i is odd. When n = 4, IrrC is Gp-invariant if and only if it is either (0¯, 0¯, 0¯, 2¯),
(2¯, 2¯, 0¯, 2¯), (1¯, 3¯, 0¯, 2¯) or if it lies in
S(2,1,1)(Vi) =
 {(Vi, 0¯, 2¯) if i is odd{(Vi, 0¯, 0¯), (Vi, 2¯, 2¯), (Vi, 1¯, 3¯)} if i is even
or if IrrC is of the form (Vi, Vj) where i+ j is odd. Therefore, N2 = m+2, N3 = 2m+5
and N4 = 3 + 4(m+ 1) + (m+ 1)
2 = (m+ 2)(m+ 4).
(ii) Reduction I∗2m: Now we assume that E has non-split reduction of type I
∗
2m, m ≥
1. The multiplicity-2 components of Emink are V−1, V0, V1, . . . , V2m−1 where δ2(Vi) = (0¯, 0¯)
if i is odd and it is equal to (1¯, 1¯) otherwise, see Corollary 6.2.3.
Assume that ψE(P ) = (0¯, 0¯). When n = 2, IrrC is Gp-invariant if and only if it is
either ((0¯, 0¯), (0¯, 0¯)), ((1¯, 1¯), (1¯, 1¯)), or Vi where i is odd. When n = 3, IrrC is Gp-invariant
if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯)), ((0¯, 0¯), (1¯, 1¯), (1¯, 1¯)), ((0¯, 1¯), (1¯, 0¯), (1¯, 1¯)),
((0¯, 0¯), Vi) where i is odd, or ((1¯, 1¯), Vi) where i is even.
When n = 4, IrrC is Gp-invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯), (0¯, 0¯)),
((0¯, 0¯), (0¯, 0¯), (1¯, 1¯), (1¯, 1¯)), ((0¯, 0¯), (0¯, 1¯), (1¯, 0¯), (1¯, 1¯)), ((0¯, 1¯), (1¯, 0¯), (0¯, 1¯), (1¯, 0¯)),
((1¯, 1¯), (1¯, 1¯), (1¯, 1¯), (1¯, 1¯)), or if it lies in
S(2,1,1)(Vi) =
 {(Vi, (0¯, 0¯), (0¯, 0¯)), (Vi, (1¯, 1¯), (1¯, 1¯)) if i is odd{(Vi, (0¯, 0¯), (1¯, 1¯)), (Vi, (0¯, 1¯), (1¯, 0¯))} if i is even
or if IrrC is of the form (Vi, Vj) where i+ j is even. Therefore, N2 = m+3, N3 = 2m+4
and N4 = 5 + 2(2m+ 1) + (m+ 1)
2 = (m+ 2)(m+ 4).
Now assume that ψE(P ) = (1¯, 1¯). When n = 2, IrrC is Gp-invariant if and only if it is
either ((0¯, 0¯), (1¯, 1¯)), ((0¯, 1¯), (1¯, 0¯)), or Vi where i is even. When n = 3, IrrC is Gp-invariant
if and only if it is either ((0¯, 0¯), (0¯, 0¯), (1¯, 1¯)), ((1¯, 1¯), (1¯, 1¯), (1¯, 1¯)), ((0¯, 0¯), (0¯, 1¯), (1¯, 0¯)),
((0¯, 0¯), Vi) where i is even, or ((1¯, 1¯), Vi) where i is odd. When n = 4, IrrC is Gp-
invariant if and only if it is either ((0¯, 0¯), (0¯, 0¯), (0¯, 0¯), (1¯, 1¯)), ((0¯, 0¯), (0¯, 0¯), (0¯, 1¯), (1¯, 0¯)),
((1¯, 1¯), (1¯, 1¯), (0¯, 0¯), (1¯, 1¯)), ((1¯, 1¯), (1¯, 1¯), (0¯, 1¯), (1¯, 0¯)), or if it lies in
S(2,1,1)(Vi) =
 {(Vi, (0¯, 0¯), (1¯, 1¯)), (Vi, (0¯, 1¯), (1¯, 0¯)) if i is odd{(Vi, (0¯, 0¯), (0¯, 0¯)), (Vi, (1¯, 1¯), (1¯, 1¯))} if i is even
or if IrrC is of the form (Vi, Vj) where i+ j is odd. Therefore, N2 = m+2, N3 = 2m+4
and N4 = 4 + 2(2m+ 1) +m(m+ 1) = (m+ 2)(m+ 3).
Reduction IV∗: If E has non-split reduction of type IV∗, then cp = 1. More pre-
cisely, EminFp consists of three irreducible components Γ,Θ,Λ of multiplicities 1, 2, 3 re-
spectively. The group Gp switches the two non-identity multiplicity-1 components, and
80
it switches the two multiplicity-2 components Θ′,Θ′′ which are not defined over Fp.
Moreover, δ2(Θ) = 0¯, δ2(Θ
′) + δ2(Θ′′) = 0¯, and δ3(Λ) = 0¯, see Proposition 6.2.5. When
n = 2, IrrC is Gp-invariant if and only if it consists of either (0¯, 0¯), (1¯, 2¯) or Θ. When
n = 3, IrrC is Gp-invariant if and only if it is either (0¯, 0¯, 0¯), (0¯, 1¯, 2¯), (0¯,Θ) or Λ. When
n = 4, IrrC is Gp-invariant if and only if it is either (0¯, 0¯, 0¯, 0¯), (0¯, 0¯, 1¯, 2¯), (1¯, 2¯, 1¯, 2¯),
(0¯, 0¯,Θ), (1¯, 2¯,Θ), (Θ,Θ), (Θ′,Θ′′) or (0¯,Λ). Hence N2 = 3, N3 = 4 and N4 = 8. 2
8.2 Proof of Theorem 8.1.1
Now we prove one of the implications of Theorem 8.1.1.
Proposition 8.2.1. Let C be a smooth curve over Qp defined by an integral genus one
equation of degree n with C(Qp) 6= ∅. Let Emin → SpecZunp be the minimal proper regular
model of the Jacobian elliptic curve E. Let (C → SpecZunp , α) be a minimal degree-n-
model for C → Pn−1Qp . If (C, α) is SpecZunp -isomorphic to a minimal degree-n-model
defined over Zp, then IrrC is Gp-invariant.
Proof: Assume that (C, α) is isomorphic to a degree-n-model (C ′ → SpecZp, α′). Set
β := α′−1α. It follows that if Γ is an irreducible component of C ′Fp , then β∗(Γ) is an
irreducible component of CFp with the same degree and multiplicity as Γ, see Theorem
5.1.4, hence IrrC′ coincides with IrrC . Now we will show that IrrC′ is Gp-invariant.
The model C ′ is obtained from Emin → SpecZunp by contraction via a divisor D′.
Moreover, H0(Emin,OEmin(mD′)), m ≥ 1, is a free finitely generated Zp-module. Hence
the divisors D′|C′Qunp and D
′|EminFp are Qp-rational and Fp-rational respectively. Since D
′
intersects irreducible components of IrrC′ and no other components in EminFp , it follows
that if Γ ∈ IrrC′ , and hence there is an x ∈ (D′|EminFp ) ∩ Γ 6= ∅, then Γ
σ ∈ IrrC′ for every
σ ∈ Gp, because xσ ∈ (D′|EminFp )
σ ∩ Γσ = (D′|EminFp ) ∩ Γ
σ. 2
The following Theorem is the main ingredient to proceed with the proof of the rest
of Theorem 8.1.1.
Theorem 8.2.2. Let E/Qp be an elliptic curve with minimal proper regular model
Emin → SpecZp. Let P ∈ E(Qp). Let n ∈ {1, 2, 3, 4}. Let (Γ1, . . . ,Γm) be a tuple of
irreducible components of EminFp satisfying:
(i) The tuple (Γ1, . . . ,Γm) is Gp-invariant.
(ii)
∑m
i=1multFp Γi = n.
81
(iii)
∑m
i=1 δmi(Γi) = ψE(P ) where mi = multFp Γi.
Then there exists a divisor D on Emin → SpecZunp such that:
(i) (D|E)σ = D|E for every σ ∈ Gal(Qunp /Qp).
(ii) D.Γi = di.multFp Γi, where di is the number of times Γi appears in (Γ1, . . . ,Γm).
(iii) D|E ∼ (n− 1).0 + P.
Remark 8.2.3. Notice that the invariance of the tuple of components under Galois
action in Theorem 8.2.2, condition (i), was always satisfied when the residue field was
algebraically closed, see Lemma 7.2.2.
Now we need a few lemmas to prove Theorem 8.2.2.
Lemma 8.2.4. Let p be a prime and d an integer such that gcd(d, p) = 1. Let K be
a finite extension of Qp. Let E/K be an elliptic curve. Assume that E/K has additive
reduction. Then the group E0(K) is divisible by d.
Proof: Let k be the residue field of K. Recall that E1(K) = {P ∈ E(K) : P˜ = 0˜E}.
The group E0(K)/E1(K) is isomorphic to k+ because E has additive reduction. In
particular, E0(K)/E1(K) is divisible by d because (d, p) = 1. But the group E1(K)
is uniquely divisible by d, see ([17], Chapter 14, Corollary 1.3). Therefore, E0(K) is
divisible by d. 2
We know that E0(Qunp )/dE0(Qunp ) = 0, where (d, p) = 1, whatever the reduction
type of E is, see ([27], Chapter VII, Exercise 7.8).
If E/Qp has non-split reduction type, then there exists a finite unramified extension
K over which E has split reduction. This field extension is quadratic except possibly
when E has non-split reduction of type I∗0, then it is either quadratic or cubic.
Let d = [K : Qp] ∈ {2, 3}. We define the norm map NormK/Qp to be NormK/Qp :
E(K)→ E(Qp); Q 7→
∑d
i=1Q
σi , where Gal(K/Qp) = 〈σ〉.
Lemma 8.2.5. Let p ≥ 5 be a prime. Assume that E/Qp has non-split reduction.
Let K be the smallest unramified extension over which E has split reduction. Then
NormK/Qp : E
0(K)→ E0(Qp) is surjective.
Proof: We first treat the additive case. So assume that E has non-split additive
reduction. Let Q ∈ E0(Qp). Since E0(Qp) is divisible by d = [K : Qp], see Lemma 8.2.4,
there is a Q′ ∈ E0(Qp) with dQ′ = Q. Now we have NormK/Qp(Q′) = dQ′ = Q.
82
Now we treat the multiplicative case. So assume E has non-split multiplicative re-
duction. We have [K : Qp] = 2. The non-singular reduction of E will be denoted by
E˜ns. Let k be the residue field of K. We denote the image of σ under the isomorphism
Gal(K/Qp) ∼= Gal(k/Fp) by σ again. Let Normk/Fp : E˜ns(k) → E˜ns(Fp); Q 7→ Q + Qσ.
Consider the following diagram.
0 // E1(K)
NormK/Qp
²²
// E0(K)
NormK/Qp
²²
// E˜ns(k)
Normk/Fp
²²
// 0
0 // E1(Qp) // E0(Qp) // E˜ns(Fp) // 0
To prove that NormK/Qp : E
0(K)→ E0(Qp) is surjective, we only need to show the
surjectivity of both NormK/Qp : E
1(K) → E1(Qp) and Normk/Fp : E˜ns(k) → E˜ns(Fp).
Let Q ∈ E1(Qp). Since E1(Qp) is divisible by 2, there is a Q′ ∈ E1(Qp) such that
2Q′ = Q. Now NormK/Qp(Q
′) = 2Q′ = Q.
Now we will show the surjectivity of Normk/Fp . The isomorphism f : E˜ns(k)
∼= k∗
induces an isomorphism E˜ns(Fp) ∼= U := {u ∈ k∗ : Normk/Fp(u) = 1}. Moreover, the
k-automorphism σ−1fσf−1 : k∗ → k∗ is u 7→ u−1, see ([20], Exercise 10.2.7). Therefore,
the map Normk/Fp induces the map k
∗ → U ; v 7→ v/vσ. Now according to Hilbert’s
Theorem 90, for u ∈ k∗ we have Normk/Fp(u) = 1 if and only if u = v/vσ for some
v ∈ k∗. Therefore, we deduce the surjectivity of Normk/Fp . 2
We need the following lemma which describes totally ramified extensions of Qp.
Lemma 8.2.6. Let L be a totally ramified extension of Qp with [L : Qp] = m, m =
2, 3, 4. Then the maximal unramified extension Lun of L is a Galois extension of Qunp
with [Lun : Qunp ] = m.
Proof: According to Corollary 3.4 of [13], since Qp is complete, Lun is a totally ramified
extension of Qunp with [Lun : Qunp ] = [L : L0], where L0 = L ∩Qunp . But since L/Qp is a
totally ramified extension, it follows that L0 = Qp. Now by virtue of Proposition 5.2.2,
since Qunp is a Henselian discrete valuation field with algebraically closed residue field,
we have Lun = Qunp ( m
√
p), and Lun/Qunp is Galois. 2
If Q = (x, y) ∈ E(Qp), then K(Q) will denote its field of definition Qp(x, y).
Proof of Theorem 8.2.2: The field k will always denote the residue field of K. The
image of σ under Gal(K/Qp) ∼= Gal(k/Fp) will be denoted by σ again. We will divide
the proof into several subcases:
(i) Assume that Λ := (Γ1, . . . ,Γm) consists of one Gp-orbit. Then we have the
following three subcases:
83
(1) If Λ consists of one irreducible component of multiplicity-n, then this component
Γ is defined over Fp and δn(Γ) = ψE(P ). We have two subcases:
– If n = 1, then set D = {P}.
– If n ≥ 2, then the reduction is additive. There exists a point x ∈ Γ defined
over Fp. According to ([20], Exercise 9.2.11 (c)), there exists a point Q′ ∈ E
such that {Q′} ∩ Γ = {x} and [K(Q′) : Qp] = n. Since x is defined over Fp,
it follows that the residue field of K(Q′) is Fp itself. Therefore, K(Q′)/Qp
is a totally ramified extension. Let L = K(Q′). According to Lemma 8.2.6,
Lun = Qunp ( n
√
p). Let Gal(Lun/Qunp ) = 〈λ〉. By the definition of δn we have
P ′ :=
∑n
i=1Q
′λi − P ∈ E0(Qp). Since E0(Qp) is divisible by n, see Lemma
8.2.4, we have P ′ = nS for some S ∈ E0(Qp). Now our divisor is D = {Q}
where Q = Q′ − S.
(2) If Λ consists of two irreducible components of multiplicity-m, then E has non-split
reduction over Qp, and this reduction splits over an unramified quadratic extension
K. Let Gal(K/Qp) = 〈σ〉. We have 2m = n, Λ = (Γ,Γσ) and δm(Γ) + δm(Γσ) =
ψE(P ). We have two further subcases to consider:
– If Γ is of multiplicity-1, then we pick x ∈ Γ to be defined over k. According to
Hensel’s Lemma, we can lift x to a point Q′ ∈ E(K). Since δm(Γ)+ δm(Γσ) =
ψE(P ), we have P
′ := Q′ + Q′σ − P ∈ E0(Qp). Lemma 8.2.5 shows that
P = S + Sσ for some S ∈ E(K). Set D = {Q} where Q = Q′ − S.
– If Γ is of multiplicity-2, then E has non-split reduction of type IV∗ over Qp.
Let x be a point on Γ defined over k. By virtue of ([20], Exercise 9.2.11
(c)), there exists a point Q′ ∈ E such that {Q′} ∩ Γ = {x} and K(Q′) is
a totally ramified extension of K with [K(Q′) : K] = 2. Let L = K(Q′),
then [Lun : Kun] = [L : L ∩ Kun] = [L : K] = 2, see ([13], Corollary 3.4).
Since K is an unramified extension of Qp, it follows that Kun = Qunp . Let
Gal(Lun/Qunp ) = 〈λ〉. If we assume that δ2(Γ) = 1¯, then by the definition of δ2
the pointQ′+Q′λ reduces on the multiplicity-1 component corresponding to 1¯.
Similarly, Q′σ+(Q′σ)λ reduces on the multiplicity-1 component corresponding
to 2¯. Therefore, we have P ′ := Q′ +Q′λ +Q′σ + (Q′σ)λ − P ∈ E0(Qp). Since
E0(Qp) is divisible by 4, see Lemma 8.2.4, we have P ′ = 4S, then we set
D = {Q} where Q = Q′ − S.
84
(3) If Λ consists of three irreducible components of multiplicity-1, then E has non-
split reduction of type I∗0 and cp = 1. The reduction splits over an unramified cubic
extension K. Let Gal(K/Qp) = 〈σ〉. We have Λ = (Γ,Γσ,Γσ2), where Γσi 6= Γσj
if i 6= j, and ∑2i=0 δ1(Γσi) = ψE(P ). Let x ∈ Γ be a point defined over k. Lift x
to a point Q′ ∈ E(K), then we have P ′ := Q′ +Q′σ + Q′σ2 − P ∈ E0(Qp). There
is a point S ∈ E(Qp) such that P ′ = 3S, see Lemma 8.2.4. Set D = {Q} where
Q = Q′ − S.
(ii) Assume that Λ := (Γ1, . . . ,Γm) consists of n Gp-orbits. Then m = n, multFp(Γi) = 1,
for each i, and
∑n
i=1 δ1(Γi) = ψE(P ). On each Γi we pick a point xi defined over Fp,
then we lift it to a point Qi ∈ E(Qp). We have P ′ :=
∑
Qi − P ∈ E0(Qp). Set
D =
∑n−1
i=1 {Qi}+ {Q′n} where Q′n = Qn − P ′.
(iii) Assume that Λ := (Γ1, . . . ,Γm) consists of two Gp-orbits and m 6= 2. Then we have
the following three subcases:
(1) Each Gp-orbit consists of one irreducible component. We have two further subcases:
– Λ = (Θ,Γ) where multFp(Θ) = n − 1 and multFp(Γ) = 1. Pick x ∈ Γ to be
defined over Fp, then we lift it to P ′ ∈ E(Qp). According to (i)-(1), there
is a divisor D′ on Emin → SpecZp such that D′|E is Qp-rational, D′|E ∼
(n− 2).0 + (P − P ′), and D′.Θ = n− 1. Set D = D′ + {P ′}.
– Λ = (Θ1,Θ2) where multFp(Θi) = 2. Let P1 ∈ E(Qp) be such that δ2(Θ1) =
ψE(P1). Let P2 = P−P1. According to (i)-(1), there are two divisorsD1, D2 on
Emin → SpecZp such that Di|E is Qp-rational, Di|E ∼ 0+Pi, and Di.Θi = 2,
where i ∈ {1, 2}. Set D = D1 +D2.
(2) One of the Gp-orbits consists of two components exactly. Then the reduction of E
splits over K where Gal(K/Qp) = {1, σ}. We have further two subcases:
– Λ = (Γ1,Γ
σ
1 ,Γ2,Γ
σ
2 ) where multFp Γi = 1 and Γi 6= Γσi , where i ∈ {1, 2}.
Let P1 ∈ E(Qp) be such that δ1(Γ1) + δ1(Γσ1 ) = ψE(P1). Let P2 = P − P1.
According to (i)-(2), there are two divisors D1, D2 on E
min → SpecZunp such
that Di|E is Qp-rational, Di|E ∼ 0 + Pi, and Di.Γi = Di.Γσi = 1, where
i ∈ {1, 2}. Set D = D1 +D2.
– Λ = (Θ,Γ,Γσ) where m := multFp(Θ) ∈ {1, 2} and Γ is a multiplicity-1
component such that Γ 6= Γσ. Let P1 ∈ E(Qp) be such that δ2(Θ) = ψE(P1).
Let P2 = P−P1. According to (i)-(1) and (i)-(2), there are two divisorsD1, D2
on Emin → SpecZunp such that Di|E is Qp-rational, D1|E ∼ (m − 1).0 + P1,
D1.Θ = m, D2|E ∼ 0 + P2 and D2.Γ = D2.Γσ = 1. Set D = D1 +D2.
85
(3) One of the Gp-orbits consists of three components exactly. Then E has non-split
reduction of type I∗0, and the reduction splits overK where Gal(K/Qp) = {1, σ, σ2}.
Moreover, we have Λ = (Γ,Γσ,Γσ
2
,Γ′) where multFp Γ
′ = 1 and Γσ
i 6= Γσj if i 6= j.
Let P ′ ∈ E(Qp) be such that δ1(Γ′) = ψE(P ′). According to (i)-(3), there is a
divisorsD′ on Emin → SpecZunp such thatD′|E isQp-rational,D′|E ∼ 2.0+(P−P ′)
and D′.Γσ
i
= 1. Set D = D′ + {P ′}.
(iv) Assume that Λ consists of three Gp-orbits andm 6= 3. We have two further subcases:
(1) Each Gp-orbit consists of one component, and hence Λ = (Θ,Γ1,Γ2), where multFp(Θ) =
2 and multFp(Γi) = 1. Let Pi ∈ E(Qp) be such that δ1(Γi) = ψE(Pi), i ∈
{1, 2}. Let P ′ := P − P1 − P2. According to (i)-(1), there is a divisor D′ on
Emin → SpecZp such that D′|E is Qp-rational, D′|E ∼ 0 + P ′, and D′.Θ = 2. Set
D = D′ + {P1}+ {P2}.
(2) One of the Gp-orbits consists of two components. Then the reduction of E splits
over K, where Gal(K/Qp) = {1, σ}, and Λ = (Γ,Γσ,Γ1,Γ2), where Γ 6= Γσ and
multFp(Γi) = 1. Let Pi ∈ E(Qp) be such that δ1(Γi) = ψE(Pi), i ∈ {1, 2}. Let
P ′ := P −P1−P2. According to (i)-(2), there is a divisor D′ on Emin → SpecZunp
such that D′|E is Qp-rational, D′|E ∼ 0 + P ′, and D′.Γ = D′.Γσ = 1. Set D =
D′ + {P1}+ {P2}.
2
Now Theorem 8.1.1 follows as a direct consequence of Theorem 8.2.2.
Proof of Theorem 8.1.1: Proposition 8.2.1 shows that if C is isomorphic to a
minimal degree-n-model defined over Zp, then IrrC is Gp-invariant. So we assume that
the tuple IrrC is Gp-invariant. The model C is obtained from Emin by contraction via a
divisor D, see Theorem 4.0.1. We want to construct a divisor D′ such that OEmin(D) ∼=
OEmin(D′) and H0(Emin,OEmin(mD′)), m ≥ 1, is a finitely generated free Zp-module.
Therefore, C ′ = Proj(⊕∞m=0H0(Emin,OEmin(mD′))) → SpecZp is a minimal degree-n-
model isomorphic to C, see Theorem 4.2.3.
The tuple IrrC = (Γ1, . . . ,Γm) satisfies the conditions of Theorem 8.2.2, for the
condition
∑m
i=1 δmi(Γi) = ψE(P ) we follow the same argument as in the proof of Lemma
7.2.1. Therefore, there exists a divisor D′ on Emin → SpecZunp such that (D′|E)σ = D′|E
for every σ ∈ Gal(Qunp /QP ), D′.Γi = di.multFp Γi, where di is the number of iterations
of Γi in the tuple (Γ1, . . . ,Γm), and D
′|E ∼ (n − 1).0 + P. It is clear that D|E ∼ D′|E
and D|EminFp ∼ D
′|EminFp , hence OEmin(D)
∼= OEmin(D′), see ([20], Exercise 9.1.13). 2
86
8.3 Counting global models
Let n ∈ {2, 3, 4}. Now we are in a place to find the number N of minimal global degree-
n-models for a smooth curve C → Pn−1Q defined by a genus one equation of degree n
such that C(Qp) 6= ∅ for every prime p. In the previous two sections we managed to
find the number Np of minimal degree-n-models for C → Pn−1Qp , at each prime p ≥ 5. In
this section, we relate N to the Np’s using Chinese Remainder Theorem.
If m ∈ Z, then we set
P (m) = {2, 3, p | p ≥ 5 is a prime , p2 | m}.
The main result of this chapter is the following Theorem.
Theorem 8.3.1. Let C → Pn−1Q be a smooth curve defined by a genus one equation of
degree n. Assume that C(Qp) 6= ∅ for every prime p. Let E/Q be the Jacobian elliptic
curve of C and let ∆ be its minimal discriminant. Let N and Np denote the numbers of
minimal global degree-n-models for C → Pn−1Q , up to isomorphism, and minimal degree-
n-models for C → Pn−1Qp , up to isomorphism, respectively. Then
N =
∏
p∈P (∆)
Np.
To prove Theorem 8.3.1 we will show that the map λ defined by
{minimal global degree-n-models for C/Q} −→
∏
p∈P (∆)
{minimal degree-n-models for C/Qp}
(C, α) 7→ ((C, α), . . . , (C, α))
is a bijection. Notice that the above two sets of degree-n-models are defined up to
isomorphism. Theorem 8.3.1 follows immediately from the bijectivity of λ.
Notice that our work enables us to compute Np for each prime p ≥ 5. Before pro-
ceeding with the proof of Theorem 8.3.1 we need two Lemmas on matrices.
Lemma 8.3.2. Let A ∈ GLn(Qp)∩Matn(Zp) have coprime entries. Then there exist ma-
trices U ∈ GLn(Z) and V ∈ GLn(Zp) such that A = V DU, where D = diag(pr1 , . . . , prn−1 , 1)
and r1 ≥ . . . ≥ rn−1.
Proof: We claim that there exists a matrix B ∈ GLn(Q) ∩Matn(Z) such that V ′ :=
BA−1 ∈ GLn(Zp). Granted this claim we write the Smith Normal Form for the matrix
B, so we have B = U ′D′DU where U,U ′ ∈ GLn(Z), D′ is a diagonal matrix whose
87
entries are not divisible by p, and D = diag(pr1 , . . . , prn−1 , 1), r1 ≥ . . . ≥ rn−1. Then we
set V := V ′−1U ′D′ ∈ GLn(Zp), hence we are done.
To prove the claim, assume that A = (aij)i,j. Recall that every element in Zp can be
written uniquely as
∑
i≥0 aip
i, where ai ∈ Z satisfies 0 ≤ ai ≤ p − 1. Let m > 0 be an
integer large enough such that the matrix B = (aij mod p
m)i,j ∈ GLn(Q) ∩Matn(Z).
Now we have BA−1 ≡ idn mod pm and hence BA−1 ∈ GLn(Zp). 2
Lemma 8.3.3. Let S = {p1, . . . , pm} be a finite set of primes. Let Ui ∈ SLn(Z) and
mi > 0 be an integer, 1 ≤ i ≤ m. Then there exists U ∈ SLn(Z) such that U ≡ Ui
mod pmii for every i, 1 ≤ i ≤ m.
Proof: That is Lemma 3.2 of [14]. 2
Recall that if R is a ring, then Gn(R) is the group of transformations of the form
[µn, An], where An ∈ GLn(R), µn ∈ R∗ when n = 2, 3, and µ4 ∈ GL2(R), see §2.1.
Lemma 8.3.4. Let φ and φ′ be two minimal Gn(Qp)-equivalent genus one equations of
degree n with coefficients in Z and Zp respectively. Then φ′ is Gn(Zp)-equivalent to a
minimal genus one equation of degree n whose coefficients lie in Z.
Proof: Assume that φ′ is obtained from φ via [µn, An] in Gn(Qp). For r ∈ Q∗p, the
following transformations are identical:
[µn, An] = [r
−2µn, rAn] when n = 2, 4, and [µ3, A3] = [r−3µ3, rA3].
Therefore, we can assume that An has coprime entries in Zp. Lemma 8.3.2 allows us to
writeAn = VnDnUn where Vn ∈ GLn(Zp), Un ∈ GLn(Z), andDn = diag(pr1 , . . . , prn−1 , 1).
Similarly, we can write µ4 = ν
′
4τ4ν4 where ν
′
4 ∈ GLn(Zp), ν4 ∈ GLn(Z), and τ4 =
diag(p−m, p−n).
Let ψ be the Z-integral genus one equation obtained from φ via the transformation
[1, Un] when n = 2, 3, and via [ν4, U4] when n = 4. Then ψ lies in the same Gn(Z)-
equivalence class as φ. Let φ′′ be the genus one equation obtained from ψ via the
transformation [µ′n, Dn], where µ
′
2 = (detD2)
−2, µ′3 = (detD3)
−1, and µ′4 = τ4. It is
clear that φ′′ is Gn(Zp)-equivalent to φ′. We claim that φ′′ has coefficients in Z. If it is
not the case, then some of the coefficients of the polynomials defining φ′′ would lie in
1
pk
Z ⊂ 1
pk
Zp for some k > 0. But φ′′ is obtained from φ′ via [ωn, V −1n ], where ωn ∈ Z∗p
when n = 2, 3, and ω4 ∈ GL2(Zp), and since φ′ is Zp-integral, it follows that φ′′ should
be Zp-integral, which is a contradiction. 2
The following lemma, ([28], Chapter IV, Lemma 9.5), will be used to justify our
choice of the set of prime numbers P (∆).
88
Lemma 8.3.5. Let K be a discrete valuation field with normalised valuation ν. Let E
be an elliptic curve over K with discriminant ∆. If ν(∆) = 1, then E has reduction of
type I1.
Proof of Theorem 8.3.1: First we will show that the map λ is well defined. Let
(C1, α1) and (C2, α2) be two isomorphic minimal global degree-n-models for C → Pn−1Q .
Then α := α−12 α1 : (C1)Qp → (C2)Qp is defined by an element in Gn(Z) ↪→ Gn(Zp) for
every prime p, i.e., (C1, α1) and (C2, α2) have the same image under λ.
To show that λ is injective, let (C1, α1) and (C2, α2) be two minimal global degree-
n-models for C → Pn−1Q with the same image under λ. We need to show that (C1, α1)
and (C2, α2) are isomorphic. Let α := α−12 α1. The map α is defined by an element
[µ,A] ∈ Gn(Q). We can assume that A ∈ Matn(Z) has coprime entries. Since (C1, α1)
and (C2, α2) have the same image under λ, hence SpecZp-isomorphic for every p ∈ P (∆),
it follows that A ∈ GLn(Zp), and so p - detA. If p 6∈ P (∆), then E/Qp has either
reduction types I0 or I1, see Lemma 8.3.5. But according to Corollary 7.1.2, when E
has either reduction types I0 or I1, there is a unique degree-n-model for C → Pn−1Qp .
That means that for p 6∈ P (∆), (C1, α1) and (C2, α2) are isomorphic as degree-n-models
for C → Pn−1Qp . Hence A ∈ GLn(Zp) for every prime p, in particular p - detA. Thus
detA = ±1 and A ∈ GLn(Z).
Now we will prove the surjectivity of λ. We will assume without loss of generality
that the defining genus one equation φ of C has coefficients in Z and that the associated
discriminant is everywhere minimal.
Let P (∆) = {p1, . . . , pm} where m ≥ 2. Let (Ci → SpecZpi , αi), where 1 ≤ i ≤ m,
be a minimal degree-n-model for C → Pn−1Qpi . We want to construct a minimal global
degree-n-model (C, α) for C → Pn−1Q such that α−1αi : (Ci)Qpi → C is defined by an
element in Gn(Zpi) for each i. Let φi be the defining genus one equation of Ci.
By virtue of Lemma 8.3.4, φi is GLn(Zpi)-equivalent to a genus one equation φ′i with
coefficients in Z. In fact, according to the proof of Lemma 8.3.4, φ′i is obtained from φ via
[µi, DiUi] where Di = diag(p
ri,1
i , . . . , p
ri,n−1
i , 1), Ui ∈ GLn(Z), and µi is a scaling element.
In fact, we can assume that Ui ∈ SLn(Z) as if detUi = −1, then we replace φ′i by the
Gn(Z)-equivalent genus one equation obtained by acting on φ′i by Vi = diag(−1, . . . , 1),
and we replace Ui by UiVi.
According to Lemma 8.3.3, given integers mi > 0, there exists a matrix U ∈ SLn(Z)
such that U ≡ Ui mod pmii for every i. We note that
∏m
i=1DiUU
−1
j D
−1
j ≡
∏
i 6=j Di
mod p
mj
j . Therefore, the genus one equation ψ obtained from φ via the transformation
[
∏m
i=1 µi,
∏m
i=1DiU ] is Gn(Zpj)-equivalent to φ′j, for every j.
Now we want to show that ψ is Z-integral. This will imply that ψ defines a mini-
mal global degree-n-model for C → Pn−1Q which is SpecZpj -isomorphic to (Cj, αj), for
89
every j. Hence we will be done with the surjectivity. Assume on the contrary that
ψ is not Z-integral. Since ψ is obtained from the Z-integral genus one equation φ via
[
∏m
i=1 µi,
∏m
i=1DiU ], it follows that some of the coefficients of the defining polynomials
of ψ lie in 1
b
Z, b = pl11 p
l2
2 . . . p
lm
m , where li ≥ 0 and lj > 0 for some j ∈ {1, . . . ,m}. We
have shown that ψ is Gn(Zpj)-equivalent to the Z-integral genus one equation φ′j. It
follows that ψ is Zpj -integral, which is a contradiction. 2
We give the following examples which show that the number of minimal global
degree-n-models, up to isomorphism, for a given smooth genus one curve agrees with
the result given by Theorem 8.3.1. Since our counting results work for minimal degree-
n-models for genus one curves defined over Qp when p ≥ 5, we choose our examples such
that the number of minimal degree-n-models defined over Zm, where m ∈ {2, 3}, is 1.
Therefore, there is no contribution of the primes 2, 3 towards the counting recipe given
in Theorem 8.3.1.
Moreover, the genus one equations of degree 2 and 3 given below are not reduced.
In fact, we moved the zeros of the defining polynomials to our favorite places to allow
applying diagonal matrices whose entries are powers of the bad primes of the Jacobian.
The calculations included in the examples below are performed using MAGMA, see [6].
Nn(T ) will denote the number of minimal degree-n-models when the reduction of the
Jacobian is of type T , see Corollary 8.1.4.
Example 8.3.6. The elliptic curve E : y2 + xy = x3 − x2 + 6603008 x − 1118312959
has bad primes 5, 7, and 11. E has reduction of types II∗, non-split I7, and III∗ at
these primes respectively. Consider the following minimal global genus one equation φ2
of degree 2.
y2 = f(x, z) = 820018280652573365 x4 + 405939973623867606 x3z +
75358348438862775 x2z2 + 6217537401171250 xz3 + 192369718165625z4.
The equation φ2 defines an everywhere locally soluble element C in the 2-Selmer group
of E. Let C be the minimal global degree-2-model defined by φ2.
We claim that C → SpecZ2 is the unique minimal degree-2-model for C. Assume on
the contrary that there is another minimal degree-2-model for C/Q2 defined by a genus
one equation φ′. The equation φ′ is obtained from φ via an element (α,A) ∈ G2(Q2).
Smith Normal Forms for matrices allow us to write A = V BU , where U, V ∈ GL2(Z2)
and B is a diagonal matrix whose entries are powers of 2. Therefore, f(x, z) mod 2 has
at least one zero, and the matrix U will move this zero to either (0, 1) or (1, 0). But
f(x, z) ≡ (x2 + xz + z2)2 mod 2, in particular f(x, z) has no zeros over F2 which is a
contradiction. We deduce that the number of minimal degree-2-models for C/Q2 is 1.
90
The model C → SpecZ3 is smooth, and hence it is the unique degree-2-model for
C/Q3, see Corollary 5.1.5. Hence the number of minimal global degree-2-models for C
is N2(II
∗)×N2(I7)×N2(III∗) = 3× 4× 2 = 24, see Theorem 8.3.1.
The defining genus one equations of the minimal global degree-2-models for C are
obtained from φ2 via the following transformations in G2(Q).
[1/(52i × 72j × 112k), diag(5i × 7j, 11k)], where 0 ≤ i ≤ 2, 0 ≤ j ≤ 3, 0 ≤ k ≤ 1.
Example 8.3.7. Consider the elliptic curve E : y2 + xy = x3 − x2 − 617x + 5916. It
has reduction of types III∗ and I2 at its bad primes 5 and 19 respectively. The following
minimal global genus one equation φ3 of degree 3 defines an everywhere locally soluble
element C in the 3-Selmer group of E.
21686353648850 x3 + 234081254700017 x2y + 9338329782950 x2z + 842219868972245 xy2 +
67198263238095 xyz + 1340388284750 xz2 + 1010096983050575 y3 + 120889031707155 y2z +
4822691362750 yz2 + 64131409475 z3 = 0.
The minimal degree-3-model C → SpecZm, m = 2, 3, defined by φ3 is smooth. There-
fore, according to Corollary 5.1.5, this model C is the unique minimal degree-3-model for
C/Qm. Hence the number of minimal global degree-3-models for C is N3(III∗)×N3(I2) =
6×2 = 12, see Theorem 8.3.1. These models have defining genus one equations obtained
from φ3 via the following transformations in G3(Q).
[1, id3], [1/5, diag(5, 1, 1)], [1/5, diag(1, 5, 1)],
[1/25, diag(5, 5, 1)], [1/25, diag(5, 1, 5)], [1/25, diag(1, 25, 1)],
[1/19, diag(1, 1, 19)], [1/95, diag(5, 1, 19)], [1/95, diag(1, 5, 19)],
[1/475, diag(5, 5, 19)], [1/475, diag(5, 1, 95)], [1/475, diag(1, 25, 19)].
Example 8.3.8. Let E : y2+xy+ y = x3− 4 x− 3. The curve C : y2 = −3 x4+2 x3+
7 x2− 2 x− 3 represents an element in the 2-Selmer group of E. A second 2-descent on
C gives the following minimal global genus one equation φ4 of degree 4.
x21 − x1x3 − x22 + x2x4 + x23 = 0,
x1x4 + x
2
2 + x2x3 − x2x4 + x23 − x3x4 = 0.
The equation φ4 defines a smooth genus one curve C4/Q. The discriminant ∆ of E is
185. Therefore, we have P (∆) = {2, 3}. But the equation φ4 defines a smooth minimal
degree-4-model C → SpecZm, m = 2, 3. Therefore, according to Corollary 5.1.5, this
model C is the unique minimal degree-4-model for C4/Qm. Hence the minimal global
degree-4-model C → SpecZ defined by φ4 is unique, see Theorem 8.3.1.
91
Appendix A
Insoluble degree-n-models
In this appendix K will denote a complete discrete valuation field with ring of integers
OK and normalised valuation ν. We will fix a uniformiser t and write k for OK/tOK .
We will assume that k is algebraically closed and that char k = p 6= 2, 3. We set S =
SpecOK .
We count the number of minimal degree-n-models for a curve C given by a genus
one equation of degree n when C(K) = ∅.
A.1 Special fibers
Let C be a smooth genus one curve over K given by a minimal genus one equation φ of
degree n = 2, 3, 4, with C(K) = ∅. Let C be a degree-n-model for C → Pn−1K . We start
by classifying the possibilities of the special fiber Ck.
Proposition A.1.1. Let C be a smooth projective curve over K. Then C(K) 6= ∅ if
and only if C admits a model C over S whose special fiber Ck contains an irreducible
component of multiplicity 1.
Proof: See ([20], Exercise 10.1.5). 2
Corollary A.1.2. Let C be a smooth curve over K given by a genus one equation of
degree n ≥ 2. Assume that C(K) = ∅. Let C be a degree-n-model for C → Pn−1K . Then
Ck is
(i) if n = 2, a double line
(ii) if n = 3, a triple line
92
(iii) if n = 4, either a double conic, two double lines, or a quadruple line.
Proof: We consider the possibilities for Ck given in §3.2. Since C(K) = ∅, Ck must not
contain multiplicity-1 components, see Proposition A.1.1. The only such forms of the
special fiber are the ones given in the statement of the corollary. 2
Let E be an elliptic curve over K with reduction type T , where
T ∈ {Im, I∗m, m ≥ 0, II, II∗, III, III∗, IV, IV∗}.
If l ≥ 0 is an integer, then we denote by lT the new type obtained from T by multiplying
all the multiplicities of T by l.
We will state some facts which will help us determine the reduction type of C when
C(K) = ∅.
Theorem A.1.3. Let X be a smooth projective curve of genus 1 over K and let E be
its Jacobian. Let Xmin → S and Emin → S be the minimal proper regular models of X
and E respectively. Let m denote the order of the element of H1(K,E) corresponding
to the torsor X. If T denotes the type of E, then X is of type mT .
Proof: See ([21], Theorem 6.6). 2
The following proposition is Corollary 7.4 in [21].
Proposition A.1.4. Let X be a smooth projective curve of genus 1 over K with minimal
proper regular model Xmin. Assume that the Jacobian E of X has additive reduction.
Let Γ1, ...,Γn be the irreducible components of X
min
k of respective multiplicities r1, ..., rn.
If r := gcd(r1, ..., rn) > 1, then r = p
s for some s ≥ 1 where p = char k.
Theorem A.1.5. Let K be a complete discrete valuation field with ring of integers
OK and algebraically closed residue field k with char k = p 6= 2, 3. Let C be a smooth
genus one curve given by a minimal genus one equation φ of degree n = 2, 3, 4. Assume
moreover that C(K) = ∅. Then the Jacobian E of C has reduction type Im, m ≥ 0.
Moreover, the minimal proper regular model Cmin of C has special fiber of type
(i) 2Im if φ is of degree 2,
(ii) 3Im if φ is of degree 3,
(iii) either 2Im or 4Im if φ is of degree 4.
93
Proof: Theorem A.1.3 implies that Cmin has reduction type lT where l is the order of
the element of H1(K,E) corresponding to the torsor C and T is the reduction type of
E. The order of an element of H1(K,E) is equal to the greatest common divisor of the
degrees of K-rational divisors on C. Since C(K) = ∅, the order l of C in H1(K,E) is n
when n = 2, 3, and it is either 2 or 4 when n = 4.
Now we want to find T. Assume on the contrary that E has additive reduction. Then
according to Proposition A.1.4, since l is the greatest common divisor of the irreducible
components of Cmink because C
min
k is of the form lT, we have l = p
s for some s > 1,
which is a contradiction as p 6= 2, 3. Therefore, the reduction type T of E is Im, m ≥ 0.
2
A.2 Insoluble degree-n-models, n = 2, 3
Recall that if f(x1, . . . , xm) ∈ OK [x1, . . . , xm], then we write fi(x1, . . . , xm) = f(x1, . . . , xm)/ti.
We start by describing the special fiber of a minimal degree-n-model for a smooth genus
one curve C → Pn−1K , for n = 2, 3, where C(K) = ∅.
Lemma A.2.1. Let φ be a minimal genus one equation of degree n = 2, 3. Assume that
φ defines a smooth curve C over K with C(K) = ∅. Let C be the minimal degree-n-
model for C → Pn−1K defined by φ. Assume that Ck contains the multiplicity-n irreducible
component {y = 0}. Then
(i) if φ : y2 = f(x, z) where ν(f) = 1, then f1(x, z) = (α1x − α2z)4 mod t for some
αi ∈ k,
(ii) if φ : by3 + f(x, z)y2 + g(x, z)y + h(x, z) where ν(b) = 0, ν(f), ν(g) ≥ 1, ν(h) = 1,
then h1(x, z) = (α1x− α2z)3 mod t for some αi ∈ k.
Proof: (i) If f1(x, z) mod t contains a linear factor, i.e., we can assume f1(x, z) =
xf ′(x, z) mod t where x - f ′(x, z), then Hensel’s Lemma allows us to lift 0 to x0 ∈ OK
with f1(x0, 1) = 0. Therefore, (x0, 0, 1) ∈ C(K), a contradiction.
Assume f(x, 1) = ax4+ bx3+ cx2+ dx+ e. If f1(x, z) mod t contains two quadratic
factors, i.e., we can assume f1(x, z) = x
2z2 mod t, then t2 | a, b, d, e, t || c. Therefore,
ν(c4) = 2, ν(c6) = 3 where c4, c6 are the invariants associated to φ, see §2.1. That
contradicts the fact that the Jacobian elliptic curve y2 = x3−27c4x−54c6 has reduction
of type Im, m ≥ 0, see Theorem A.1.5.
(ii) If h1(x, z) mod t contains a linear factor, i.e., we can assume h1(x, z) = xh
′(x, z)
mod t where x - h′(x, z), then Hensel’s Lemma allows us to lift 0 to x0 ∈ OK with
h1(x0, 1) = 0. Therefore, (x0 : 0 : 1) ∈ C(K), which is a contradiction. 2
94
Remark A.2.2. If C is given by a minimal genus one equation y2 = f(x, z) = ax4 +
bx3z+ cx2z2+dxz3+ ez4 and C(K) = ∅, then we can assume that f1(x, z) = x4 mod t,
see Lemma A.2.1 (i). Therefore, ν(a) = 1 and min{ν(b), ν(c), ν(d), ν(e)} ≥ 2. Indeed,
ν(e) ≥ 3 since otherwise f2(tx, z) 6≡ 0 mod t which contradicts Corollary A.1.2 (i).
If ν(e) ≥ 4 then ν(d) = 2 otherwise f2(tx, z) is not minimal, but then f3(tx, z) has
a linear factor which contradicts Lemma A.2.1 (i). Therefore, ν(e) = 3. Moreover, if
ν(d) = 2 then again f3(tx, z) has a linear factor.
To summarise, we can assume that ν(a) = 1,min{ν(b), ν(c)} ≥ 2, ν(d) ≥ 3, and
ν(e) = 3.
Let C be given by a minimal genus one equation F (x, y, z) = by3 + f(x, z)y2 +
g(x, z)y + h(x, z) = 0 where g(x, z) = a2x
2 +mxz + c2z
2 and h(x, z) = ax3 + a3x
2z +
c1xz
2+cz3.We will assume that C(K) = ∅, ν(b) = 0, ν(f), ν(g) ≥ 1 and ν(h) = 1. Then
according to Lemma A.2.1 (ii) we can assume that h1(x, z) = x
3 mod t and therefore
ν(a) = 1, min{ν(a3), ν(c1), ν(c)} ≥ 2 and c 6= 0.
Indeed, we have ν(c) = 2 and ν(c2) ≥ 2, since otherwise the equation F2(tx, ty, 1) = 0
would define a minimal degree-3-model for C → P2K whose special fiber is a double line
and a line which contradicts Corollary A.1.2 (ii).
A curve defined by a genus one equation with coefficients of the above valuations is
called a critical model in [11].
Theorem A.2.3. Assume that C is a smooth curve given by a minimal genus one
equation φ1 of degree n = 2, 3. Assume moreover that C(K) = ∅. Then the number of
minimal degree-n-models for C → Pn−1K is n.
Proof: n = 2 : The equation φ1 : y
2 = f(x, 1) = ax4+bx3+cx2+dx+e where t || f(x)
defines a minimal degree-2-model C1 for C. According to Remark A.2.2 we are allowed
to assume that ν(a) = 1,min{ν(b), ν(c)} ≥ 2, ν(d) ≥ 3, and ν(e) = 3. We claim that
the models (C1, id) and (C2, α2), where C2 is given by φ2 : y2 = f2(tx, 1) and hence α2 is
defined by the matrix diag(t, 1), are the only minimal degree-2-models for C → P1K . So
assume that (C3, α3) is another minimal degree-2-model for C → P1K . We will show that
C3 is isomorphic to either C1 or C2. Assume that C3 is defined by the genus one equation
φ3 : y
2 = 1
(detA)2
f(a1(x, z), a2(x, z)) where
A =
 a11 a12
a21 a22
 ,
and ai(x, z) = a1ix+a2iz, i = 1, 2, moreover we will assume that A has coprime entries.
If t | detA, then t2 | f(a1(x, z), a2(x, z)) because φ3 is integral, hence t | a1(x, z) because
95
ν(a) = 1. If t2 | detA, then t4 | f(a1(x, z), a2(x, z)) and so t | a2(x, z) because ν(e) = 3.
Therefore, if detA is divisible by a power of t which is greater than 1, then t divides
each entry of A which contradicts our assumption. Thus we can assume that if t | detA,
then t || detA.
Now assume that C3 is not isomorphic to C1, hence t || detA. We will show that
C3 must be isomorphic to C2. Writing A in a Smith Normal Form we can assume that
A = B′TB, where
B′ ∈ GL2(OK), T =
 t 0
0 1
 , and B =
 b11 b12
b21 b22
 ∈ GL2(OK).
We apply the matrix B′−1 which does not change the isomorphism class of C3. So we
can assume without loss of generality that A = TB. The defining genus one equation
φ3 : y
2 = f ′(x, z) of C3 satisfies f ′(x, z) = TBT−1.f2(tx, z). So we only need to show
that TBT−1 ∈ GL2(OK). Noting that
TBT−1 =
 b11 tb12
b21/t b22
 ,
we have to show that t | b21. Let e′ be the coefficient of the z4-term in
f ′(x, z) = f(t(b11x+ b21z/t), tb12x+ b22z)/t2.
Hence e′ = f ′(b21, b22)/t2 ≡ ab421/t2 mod OK . Since ν(a) = 1, we conclude that t | b21,
since otherwise φ3 is not integral.
n = 3 : Let φ1 be given by F (x, y, z) = by
3+ f(x, z)y2+ g(x, z)y+h(x, z) = 0 where
g(x, z) = a2x
2+mxz+c2z
2, h(x, z) = ax3+a3x
2z+c1xz
2+cz3, and ν(f), ν(g), ν(h) ≥ 1.
By virtue of Remark A.2.2, we can assume that ν(a) = 1, min{ν(a3), ν(c1), ν(c2)} ≥
2, ν(c) = 2, and c 6= 0. We claim that the models (C1, id), (C2, α2) and (C3, α3) given by
the genus one equations F (x, y, z) = 0, φ2 : F1(x, ty, z) = 0, and φ3 : F2(tx, ty, z) = 0
respectively are the only degree-3-models for C → P2K . So assume that (C4, α4) is another
degree-3-model for C → P2K and we want to prove that C4 is isomorphic to one of the
Ci’s, i = 1, 2, 3. Let C4 be given by φ4 : 1detAF (a1(x, y, z), a2(x, y, z), a3(x, y, z)) = 0
where
A =

a11 a12 a13
a21 a22 a23
a31 a32 a33
 ,
96
and ai(x, y, z) = a1ix+ a2iy + a3iz, i = 1, 2, 3, moreover we assume that A has coprime
entries. If t3 | detA, then t | a2(x, y, z) because φ4 is integral and ν(b) = 0, similarly
t | a1(x, y, z) because ν(a) = 1, and t | a3(x, y, z) because ν(c) = 2, i.e., if t3 | detA, then
that contradicts our assumption that A has coprime entries. Therefore, ν(detA) ≤ 2.
Now we assume that C4 is not isomorphic to C1, hence we can assume that A is written
as TB where
T =

tr1 0 0
0 tr2 0
0 0 1
 , B =

b11 b12 b13
b21 b22 b23
b31 b32 b33
 ∈ GL3(OK),
and r1 ≥ r2, r1 + r2 ≤ 2. We first eliminate the possibility of r1 = 2, r2 = 0. That is
because if we have a genus one equation of degree 3 which is K-equivalent to φ1 and
given by
F ′(x, y, z) = Ax3 +By3 + Cz3 + A2X2Y + A3X2Z +B1Y 2X
+ B3Y
2Z + C1Z
2X + C2Z
2Y +MXY Z
such that F ′2(t
2x, y, z) ∈ OK [x, y, z], then we have t2 | B,C,B3, C2. But the model
corresponding to the integral genus one equation F ′2(t
2x, y, z) = 0 has a triple line special
fiber, see Corollary A.1.2, and the reduction of F ′2(t
2x, y, z) mod t contains no y3, z3-
terms, therefore t | A2, A3, B1, C1,M. Now the equation F ′(x, y, z) = 0 is not minimal
as we can minimise it using the transformation F ′2(tx, y, z). So either r1 = 1, r2 = 0 or
r1 = r2 = 1.
We will show that if r1 = r2 = 1, then C4 is isomorphic to C3. The case when
r1 = 1, r2 = 0 is similar and we have C4 is isomorphic to C2. Let r1 = r2 = 1. We need
to show that TBT−1 ∈ GL3(OK). We notice that for r1 = r2 = 1 we have
TBT−1 =

b11 b12 tb13
b21 b22 tb23
b31/t b32/t b33
 ,
so we have to prove that t | b31, b32. C4 is defined by the equation
TBT−1.F2(tx, ty, z) = F (t(b11x+b21y+b31z/t), t(b12x+b22y+b32z/t), tb13x+tb23y+b33z)/t2 = 0.
In the above polynomial the coefficient of z3 is c′ = F (b31, b32, b33)/t2. If ν(b32) = 0, then
since ν(b) = 0, we would have ν(c′) = −2, which contradicts the integrality of the genus
one equation above. Hence ν(b32) ≥ 1. Now if ν(b31) = 0, then since ν(a) = 1, we would
have ν(c′) = −1, which is again a contradiction. Therefore, t | b31, b32. 2
97
A.3 Insoluble degree-4-models
Let φ be a genus one equation of degree 4 given by {F (x1, . . . , x4) = G(x1, . . . , x4) = 0}
where F and G are given by the following polynomials respectively
a1x
2
1 + a2x1x2 + a3x1x3 + a4x1x4 + a5x
2
2 + a6x2x3 + a7x2x4 + a8x
2
3 + a9x3x4 + a10x
2
4,
b1x
2
1 + b2x1x2 + b3x1x3 + b4x1x4 + b5x
2
2 + b6x2x3 + b7x2x4 + b8x
2
3 + b9x3x4 + b10x
2
4.
(A.1)
Assume that φ defines a smooth curve C with C(K) = ∅. Then Corollary A.1.2 (iii)
implies that the special fiber of any degree-4-model for C → P3K is either a double conic,
two double lines, or a quadruple line.
In Chapter 3 we introduced conditions for a degree-4-model for C to be normal and
we determined its singular locus. We are going to investigate the singular locus of a
minimal degree-4-model for C → P3K under the assumption that C(K) = ∅. Then we
show that unlike the case when n = 2, 3, the number of minimal degree-4-models for
C → P3K can be arbitrarily large.
We start by stating the following version of Hensel’s Lemma.
Lemma A.3.1. Let f1, . . . , fr ∈ OK [x1, . . . , xn], r ≤ n. Let x ∈ OnK be such that fi(x) =
0 mod t for every i and
rank(
∂fi
∂xj
(x) mod t) ≥ r.
Then there exists a y ∈ OnK such that y ≡ x mod t and fi(y) = 0 for every i.
Again fi will denote f/t
i.
Lemma A.3.2. Let C be a smooth genus one curve over K defined by an integral genus
one equation φ : F = G = 0 of degree 4 as in equation (A.1). Assume that C(K) = ∅.
Assume moreover that φ defines a minimal degree-4-model C for C → P3K .
(i) Assume that Ck is a double conic with F˜ = x21 and G˜ = x22+x3x4. Then F˜1(0, x2x4,−x22, x24) =
(α1x2 − α2x4)4 where αi ∈ k.
(ii) Assume that Ck is two double lines given by {x1 = x2 = 0} and {x1 = x4 = 0}.
Then both F˜1(0, 0, x3, x4) and F˜1(0, x2, x3, 0) are squares.
(iii) Assume that Ck is a quadruple line given by F˜ = x21 and G˜ = x22 + x1x3. Then
F˜1(0, 0, x3, x4) is a square.
98
(iv) Assume that Ck is a quadruple line given by F˜ = x21 and G˜ = x22. Then both
F˜1(0, 0, x3, x4) and G˜1(0, 0, x3, x4) are squares.
Proof: Set
f(x2, x3) = F1(0, x2, x3, 1) = a5,1x
2
2 + a6,1x2x3 + a7,1x2 + a8,1x
2
3 + a9,1x3 + a10,1,
f ′(x2, x3) = G(0, x2, x3, 1) = b5x22 + b6x2x3 + b7x2 + b8x
2
3 + b9x3 + b10.
(i) Let h(x2, x4) = F (0, x2x4,−x22, x24). Assume that h1(x2, x4) mod t has a simple
factor. Using a matrix in GL4(OK) we can assume that h1(x2, x4) = x2g(x2, x4), where
x2 - g(x2, x4), therefore ν(a7) = 1 and ν(a10) ≥ 2.
we have f(0, 0) = f ′(0, 0) = 0 mod t and
J(f,f ′) :=
 ∂f∂x2 ∂f∂x3
∂f ′
∂x2
∂f ′
∂x3
 (0, 0) =
 a7,1 a9,1
b7 1
 ≡
 a˜7,1 0
0 1
 mod t.
Lemma A.3.1 implies that there are x, y ∈ OK such that f(x, y) = f ′(x, y) = 0 which
means that (0 : x : y : 1) ∈ C(K), whence a contradiction.
When a5 6= a9, the polynomial h1(x2, x4) can have two double factors mod t, in
this case we can use a matrix in GL4(OK) to assume that h1(x2, x4) = x22x24 mod t, in
particular we assume that min{ν(a6), ν(a7), ν(a8), ν(a10)} ≥ 2 and ν(a5 − a9) = 1. But
then the degree-4-model defined by the genus one equation
F1(tx1, x2, x3, x4) = G(tx1, x2, x3, x4) = 0
has special fiber with equations a5,1x
2
2 + a9,1x3x4 = l(x2, x4)
2 + x3l
′(x2, x4) = 0, where l
and l′ are linear factors. The latter special fiber contradicts either Corollary A.1.2 (iii)
or Theorem A.1.5 (iii).
(ii) We can assume that F˜ = x21 and G˜ = x2x4+µx1x3 where µ ∈ k. If F˜1(0, 0, x3, x4)
consists of two distinct linear factors, then we can assume that ν(a9) = 1 and ν(a8), ν(a10) ≥
2. Now we have
J(f,f ′) ≡
 a˜7,1 a˜9,1
1 0
 mod t.
That implies the existence of a rational point on C which is a contradiction. We follow
the same argument to prove that F˜1(0, x2, x3, 0) is a square.
(iii), (iv) Assume that F˜1(0, , 0, x3, x4) consists of two distinct linear factors. We can
use a matrix in GL4(OK) to assume that ν(a9) = 1 and ν(a8), ν(a10) ≥ 2. But then the
degree-4-model C ′ for C → P3K given by
F1(tx1, tx2, x3, x4) = G1(tx1, tx2, x3, x4) = 0
99
has a special fiber of the form a9,1x3x4 = µx1l(x3, x4) + G˜1(0, 0, x3, x4) where µ ∈ k and
l is a linear factor. Therefore, C ′k contains a multiplicity-1 component which contradicts
Corollary A.1.2 (iii). Similar argument works for the polynomial G˜1(0, 0, x3, x4) of (iv).
2
Remark A.3.3. Let C be as in Lemma A.3.2. Let Sing(C) be the singular locus of C.
If Ck is a double conic with defining equations x21 = x22 + x3x4 = 0, then after a
transformation in GL4(OK) we can assume that ν(a8) = 1 and
min{ν(a5), ν(a6), ν(a7), ν(a9), ν(a10)} ≥ 2,
see Lemma A.3.2. Moreover, Sing(C) = {(0 : 0 : 0 : 1)}, see Proposition 3.3.6 (iii).
Assume that Ck is two double lines with equations x21 = x2x4 + µx1x3 = 0. If we
assume that ν(a5) = ν(a10) = 1 and min{ν(a6), ν(a8), ν(a9)} ≥ 2, then the degree-4-
model defined by the genus one equation
F1(tx1, x2, x3, x4) = G(tx1, x2, x3, x4) = 0
has special fiber with equations a5,1x
2
2 + a7,1x2x4 + a10,1x
2
4 = x2x4 = 0, which is a
contradiction. Therefore, if Ck is two double lines given by the above equations, then
we can assume that
ν(a8) = 1, min{ν(a5), ν(a6), ν(a9), ν(a10)} ≥ 2,
and Sing(C) = {(0 : 1 : 0 : 0), (0 : 0 : 0 : 1)}, see Proposition 3.3.6 (iii).
If Ck is a quadruple line, then we have two possibilities according to Lemma A.3.2:
(i) the defining equations of Ck are x21 = x22 + x1x3 = 0 with ν(a8) = ν(b10) =
1, ν(a9), ν(a10) ≥ 2, Sing(C) = {(0 : 0 : 0 : 1)}, or ν(a10) = 1, ν(a8), ν(a9) ≥ 2,
Sing(C) = {(0 : 0 : 1 : 0)}. (ii) the defining equations of Ck are x21 = x22 = 0 with
ν(a8) = ν(b10) = 1 and min{ν(a9), ν(a10), ν(b8), ν(b9)} ≥ 2.
Now we give an example to show that the number of non-isomorphic minimal degree-
4-models for C → P3K may become arbitrarily large when C(K) = ∅.
Example A.3.4. Let i > 0 be an integer. Let φ be a minimal genus one equation of
degree 4 as in Equation (A.1) with the following coefficients valuations
100
x21 x1x2 x1x3 x1x4 = 0 ≥ i ≥ 1 ≥ i ≥ 1 ≥ 1 ≥ 1 ≥ 1
x22 x2x3 x2x4 ≥ 2i ≥ i ≥ 2i = 0 ≥ 1 ≥ 1
x23 x3x4 = 1 ≥ i ≥ 2 ≥ 2
x24 ≥ 2i = 1.
The equation φ defines a degree-4-model C for the curve C → P3K defined by the
same equation. We have C(K) = ∅, see ([11], Lemma 5.2). In addition, this genus one
equation φ is minimal, see ([11], Lemma 5.3). The special fiber is a quadruple line. We
define non-isomorphic degree-4-models (Cm, αm), 1 ≤ m ≤ i, for C → P3K , where Cm is
given by
F2m(t
mx1, x2, t
mx3, x4) = G(t
mx1, x2, t
mx3, x4) = 0.
101
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