Vol. 7, No. 9, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Vinberg's representations and arithmetic invariant theory

Jack A. Thorne

Vol. 7 (2013), No. 9, 2331–2368

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 .

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