Explicit computations of Serre's obstruction for genus 3 curves and application to optimal curves

TL;DR

This paper provides explicit methods to compute Serre's obstruction for genus 3 curves, enabling the determination of whether certain abelian threefolds are Jacobians, with applications to identifying optimal genus 3 curves.

Contribution

It introduces explicit computational techniques for Serre's obstruction in genus 3, especially for products of CM elliptic curves, and applies these to study optimal curves.

Findings

Explicit computation of Serre's obstruction for specific abelian threefolds

Determination of the existence of certain optimal genus 3 curves

Validation or refutation of conjectured curves using numerical methods

Abstract

Let k be a field of characteristic different from 2. There can be an obstruction for an indecomposable principally polarized abelian threefold (A,a) over k to be a Jacobian over k. It can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numeric results to prove or refute the existence of some optimal curves of genus 3.