Fake 2-descent on the Jacobian of a genus-3 curve

For many questions in explicit arithmetic geometry of curves, one needs detailed information on the rational points of the Jacobian of the curve. A first step is to bound the free rank of the finitely generated group that they form. For hyperelliptic curves [curves admitting a model of the form y2 = f(x)], we have fairly good methods for producing bounds, and curves of genus 2 are always hyperelliptic. Curves of genus 3 (for instance smooth plane quartics) are generally not hyperelliptic. A straightforward generalization of the standard methods to these curves would lead to infeasible computational tasks involving number fields up to degree 756. We propose a modification, which requires number fields up to degree 28 and is sometimes just about feasible.