Arithmetic invariant theory and 2-descent for plane quartic curves

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 $J\left(k\right)∕2J\left(k\right)\hookrightarrow G\left(k\right)\setminus X\left(k\right)$. 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.