#### Vol. 10, No. 7, 2016

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Arithmetic invariant theory and 2-descent for plane quartic curves

### Appendix: Tasho Kaletha

Vol. 10 (2016), No. 7, 1373–1413
##### Abstract

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

##### Keywords
arithmetic geometry, descent, invariant theory
Primary: 11D25
Secondary: 11E72
##### Milestones
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