Arithmetic invariant theory and 2-descent for plane quartic curves

Thorne, Jack

doi:10.2140/ant.2016.10.1373

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Arithmetic invariant theory and 2-descent for plane quartic curves

Jack A. Thorne

Appendix: Tasho Kaletha

Vol. 10 (2016), No. 7, 1373–1413

DOI: 10.2140/ant.2016.10.1373

Abstract

Given a smooth plane quartic curve $C$ over a field $k$ of characteristic 0, with Jacobian variety $J$ , and a marked rational point $P \in C (k)$ , we construct a reductive group $G$ and a $G$ -variety $X$ , together with an injection $J (k) ∕ 2 J (k) ↪ G (k) ∖ X (k)$ . We do this using the Mumford theta group of the divisor $2 Θ$ of $J$ , and a construction of Lurie which passes from Heisenberg groups to Lie algebras.

Keywords

arithmetic geometry, descent, invariant theory

Mathematical Subject Classification 2010

Primary: 11D25

Secondary: 11E72

Milestones

Received: 2 April 2015

Revised: 29 April 2016

Accepted: 18 July 2016

Published: 27 September 2016

Authors

Jack A. Thorne
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom

Tasho Kaletha
Department of Mathematics University of Michigan 530 Church Street Ann Arbor, MI 48109-1043 United States

Vol. 10, No. 7, 2016