Certifying nontriviality of Ceresa classes of curves

TL;DR

This paper develops an algorithm to certify the non-torsion property of Ceresa classes of algebraic curves over number fields, providing a practical tool for understanding their algebraic cycles without symmetry assumptions.

Contribution

It introduces a new algorithm for certifying non-torsion of Ceresa classes and proves its correctness under the full Sato--Tate group hypothesis.

Findings

Algorithm often certifies non-torsion of Ceresa classes.

Under Sato--Tate group assumptions, the algorithm terminates with a certificate.

Provides a method to verify nontriviality of Ceresa classes without extra symmetries.

Abstract

The Ceresa cycle is a canonical algebraic $1$-cycle on the Jacobian of an algebraic curve. We construct an algorithm which, given a curve over a number field, often provides a certificate that the Ceresa cycle is non-torsion, without relying on the presence of any additional symmetries of the curve. Under the hypothesis that the Sato--Tate group is the whole of $\operatorname*{GSp}$, we prove that if the Ceresa class (the image of the Ceresa cycle in \'{e}tale cohomology) is non-torsion, then the algorithm will eventually terminate with a certificate attesting to this fact.