Arithmetic invariant theory and 2-descent for plane quartic curves

Abstract

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