Certifying nontriviality of Ceresa classes of curves

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.