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.

💡 Research Summary

The paper revisits the classical Manin‑Drinfeld theorem, which asserts that any degree‑zero divisor supported on the cusps of a modular curve becomes torsion in its Jacobian, and re‑derives it using Elkik’s mixed Hodge‑theoretic approach. Building on this, the author first recovers Rohrlich’s extension of the theorem to Fermat curves, then pushes the method further to higher‑dimensional Fermat hypersurfaces and to higher Chow groups.

The core technical tool is the description of the Abel‑Jacobi map as an element of an Ext‑group of mixed Hodge structures (MHS). For a smooth projective variety X and a null‑homologous cycle Z supported on a divisor D, one obtains a short exact sequence of MHS \