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