Vinberg's representations and arithmetic invariant theory

Abstract

Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over $\mathbb{Q}$ as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves.

In this paper we construct for each simply laced Dynkin diagram a coregular representation $\left(G,V\right)$ and a family of algebraic curves over the geometric quotient $V∕∕G$. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of $G$. In the case of type $A_2$, we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over $\mathbb{Q}$.