Arithmetic invariant theory and 2-descent for plane quartic curves

TL;DR

This paper develops a new algebraic framework linking the 2-torsion points of a plane quartic curve's Jacobian to group actions, utilizing invariant theory and theta groups to facilitate 2-descent methods.

Contribution

It introduces a novel construction of a reductive group and variety that connect Jacobian 2-torsion points to group orbits, advancing the algebraic tools for studying quartic curves.

Findings

Established an injection from J(k)/2J(k) to G(k)ackslash X(k)

Utilized Mumford theta groups and Lurie's construction to relate Heisenberg groups to Lie algebras

Provided a new approach for 2-descent on 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 of C(k), we construct a reductive group G and a G-variety X, together with an injection J(k)/2J(k) -> G(k)\X(k). We do this using the Mumford theta group of J, and a construction of Lurie which passes from Heisenberg groups to Lie algebras.