A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic

TL;DR

This paper computes harmonic volume values for the Fermat sextic and demonstrates that a specific algebraic cycle in its Jacobian variety is not algebraically equivalent to zero, revealing new insights into algebraic cycles.

Contribution

It introduces a novel computation of harmonic volume for the Fermat sextic and proves a nontrivial algebraic cycle is not algebraically equivalent to zero.

Findings

Harmonic volume values computed for Fermat sextic

Proved a specific algebraic cycle is nontrivial in the Jacobian

Enhanced understanding of algebraic cycles in Fermat curves

Abstract

We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.