Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics

A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ 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 paper, 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-Th\'el\`ene and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Bialynicki-Birula decomposition, and a general construction of integral orbit representatives.