Generalized explicit descent and its application to curves of genus 3

TL;DR

This paper presents a unified method for computing Selmer groups applicable to genus 3 curves, extending practical computations beyond genus 2 and simplifying theoretical proofs.

Contribution

It introduces a generalized explicit descent method that is practical for genus 3 curves, unifying and extending previous approaches.

Findings

Successfully applied to some genus-3 examples with small coefficients

Simplifies the connection between computations and cohomological Selmer groups

First practical approach for Selmer group computations on genus-3 curves

Abstract

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically-defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over Q of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus-3 examples defined by polynomials with small coefficients.