On the Abel-Jacobi maps of Fermat Jacobians

TL;DR

This paper investigates the Abel-Jacobi images of Ceresa cycles on Fermat Jacobians, expressing them via hypergeometric functions and providing criteria for their non-vanishing, supported by numerical evidence.

Contribution

It introduces a new expression for Ceresa cycles on Fermat Jacobians using special hypergeometric values and establishes a non-vanishing criterion with numerical verification.

Findings

Expressed Ceresa cycles in terms of hypergeometric functions

Provided a criterion for non-vanishing of Ceresa cycles

Numerically verified the criterion for specific cases

Abstract

We study the Abel-Jacobi image of the Ceresa cycle W_k-W_k^-, where W_k is the image of the k-th symmetric product of a curve X on its Jacobian variety. For the Fermat curve of degree N, we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non-vanishing of W_k-W_k^- modulo algebraic equivalence, which is verified numerically for some N and k.