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Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics 10.17863/CAM.84919 Laga, Jef 0000000339506490 Thorne, Jack A simply laced Dynkin diagram gives rise to a family of curves over Q and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations. In this thesis, we complete Thorne's picture and show that $2$-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are an analysis of torsors on affine spaces using results of Colliot--Thelene and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives. University of Cambridge 2021-11-30 Thesis Doctoral Doctor of Philosophy (PhD) eng ERC (714405) https://www.repository.cam.ac.uk/handle/1810/337504 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/5776b6d1-2842-412d-8658-3130b62b899b/download 1a972ac7f4823e7d975817da31e441c9 https://www.rioxx.net/licenses/all-rights-reserved/ Algebraic curves Arithmetic statistics Lie algebras Rational points Geometry of numbers