Torsion cycles on Fermat varieties

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.