Open Access
2013 Vinberg's representations and arithmetic invariant theory
Jack Thorne
Algebra Number Theory 7(9): 2331-2368 (2013). DOI: 10.2140/ant.2013.7.2331


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 .


Download Citation

Jack Thorne. "Vinberg's representations and arithmetic invariant theory." Algebra Number Theory 7 (9) 2331 - 2368, 2013.


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

zbMATH: 1321.11045
MathSciNet: MR3152016
Digital Object Identifier: 10.2140/ant.2013.7.2331

Primary: 20G30
Secondary: 11E72

Keywords: arithmetic invariant theory , arithmetic of algebraic curves , Galois cohomology

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 9 • 2013
Back to Top