Vol. 7, No. 9, 2013

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Vinberg's representations and arithmetic invariant theory

Jack A. Thorne

Vol. 7 (2013), No. 9, 2331–2368
Abstract

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 .

Keywords
arithmetic invariant theory, Galois cohomology, arithmetic of algebraic curves
Mathematical Subject Classification 2010
Primary: 20G30
Secondary: 11E72
Milestones
Received: 8 November 2012
Revised: 14 February 2013
Accepted: 17 March 2013
Published: 18 December 2013
Authors
Jack A. Thorne
Department of Mathematics
Harvard University
1 Oxford Street
Cambridge, MA 02138
United States