2026-09-23T12:00:16Zhttps://api.repository.cam.ac.uk/server/oai/request

oai:www.repository.cam.ac.uk:1810/2649392024-06-26T13:47:05Zcom_1810_213747com_1810_256064col_1810_213748

On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication 10.17863/CAM.10705 Kezuka, Yukako Coates, John Henry The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\mathfrak{p}\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\infty=HK_\infty$, where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. We establish in this thesis the main conjecture for the extension $H_\infty/H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H_\infty/H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$. University of Cambridge 2017-05-30 Thesis Doctoral Doctor of Philosophy (PhD) en https://www.repository.cam.ac.uk/handle/1810/264939 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/d3d613e1-6d73-4177-b724-9148d9d83073/download afa0f3a92c3384d5fa583073ea822027 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/26dd4726-c3a5-4c55-a407-d90de98c25fb/download 87eda9de84448d1f82354d60eee3eb5f https://www.rioxx.net/licenses/all-rights-reserved/ Iwasawa theory Elliptic curves Complex Multiplication CM Birch-Swinnerton-Dyer conjecture BSD main conjecture p-adic L-function Elliptic units Quadratic twists Cubic twists L-series Gross curve L-value Rubin