Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves

TL;DR

This paper develops an algorithm to show that certain algebraic cycles, specifically Ceresa cycles, are not algebraically equivalent to zero in Jacobian varieties of specific Fermat curve quotients.

Contribution

It introduces a method to compute the trace map of harmonic volumes, demonstrating nontriviality of Ceresa cycles in these Jacobians.

Findings

Ceresa cycles are not algebraically equivalent to zero in the studied Jacobians.

Provides an explicit algorithm for detecting nontrivial algebraic cycles.

Advances understanding of algebraic cycles in Fermat curve quotients.

Abstract

We obtain the trace map image of the values of certain harmonic volumes for some quotients of Fermat curves. This provides the algorithm that the algebraic cycles called by the k-th Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties.