Twists of genus three Jacobians

TL;DR

This paper provides an arithmetic criterion to distinguish genus three Jacobians from their [-1] twists using theta nulls, linking these invariants to the curve's discriminant, and generalizes prior work on Abelian threefolds.

Contribution

It introduces a new criterion based on theta nulls to identify genus three Jacobians and relates it to the discriminant, extending previous results to a broader class of Abelian threefolds.

Findings

Criterion based on theta nulls distinguishes Jacobians from twists.

Expresses theta null product in terms of the curve's discriminant.

Generalizes previous work on isogenous Abelian threefolds.

Abstract

We give a criterion to distinguish between a genus three Jacobian and its [-1] twist in terms of the product of the 36 even theta nulls. We also express the product of the 36 theta nulls in terms of the discriminant of a genus three curve. The results are arithmetic in nature and thus add to previous work over C on the product of the even theta nulls. They generalize previous work of Ritzenthaler and Lachaud for Abelian threefolds which are (2,2,2) isogenous to a product of elliptic curves. This answers a 2003 of letter of Jean-Pierre Serre to Jaap Top.