Experiments with Ceresa classes of cyclic Fermat quotients

TL;DR

This paper presents new examples of non-hyperelliptic curves with torsion Ceresa cycles in the intermediate Jacobian, linking their properties to $L$-function values and conjectures in algebraic geometry.

Contribution

It introduces two novel non-hyperelliptic curves with torsion Ceresa cycles and explores their relation to $L$-function non-vanishing and existing conjectures.

Findings

Existence of non-hyperelliptic curves with torsion Ceresa cycles.

One example has a non-vanishing $L$-function value.

Speculation on the origin of torsion Ceresa classes based on cyclic Fermat quotients.

Abstract

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, the central value of the $L$-function of the relevant motive is non-vanishing and the Ceresa cycle is torsion in the Griffiths group, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients.