Algebra & Number Theory

Vinberg's representations and arithmetic invariant theory

Jack Thorne

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Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over 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 (G,V) and a family of algebraic curves over the geometric quotient VG. 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 A2, 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 .

Article information

Algebra Number Theory, Volume 7, Number 9 (2013), 2331-2368.

Received: 8 November 2012
Revised: 14 February 2013
Accepted: 17 March 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G30: Linear algebraic groups over global fields and their integers
Secondary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10]

arithmetic invariant theory Galois cohomology arithmetic of algebraic curves


Thorne, Jack. Vinberg's representations and arithmetic invariant theory. Algebra Number Theory 7 (2013), no. 9, 2331--2368. doi:10.2140/ant.2013.7.2331.

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