Torsion cycles on Fermat varieties

TL;DR

This paper revisits Rohrlich's generalization of Manin and Drinfeld's theorem on torsion divisors on modular curves, extending the results to higher codimensional cycles and Fermat varieties using mixed Hodge theory.

Contribution

It provides a new proof of Rohrlich's results and extends the torsion property to higher Chow cycles on Fermat varieties.

Findings

Torsion properties of divisors on Fermat curves confirmed

Extension of torsion results to higher codimension cycles

Application of mixed Hodge theory to Fermat varieties

Abstract

A theorem of Manin and Drinfeld states that any divisor of degree $0$ on the cusps of a modular curve is torsion in the Jacobian. An elegant proof of this result was provided by Elkik using mixed Hodge theory. Rohrlich proved a generalization of this to Fermat curves. In this note we reprove his results along the lines of the work of Elkik. We then use the same methods to generalize it to higher codimensional null-homologous cycles as well as higher Chow cycles on Fermat varieties.