Ceresa cycles of Fermat curves

The Ceresa cycle of a smooth complex projective curve of positive genus is the formal difference of a copy of the curve in its Jacobian (via choosing a base point) and the image of that copy under the inversion map of the Jacobian. Ceresa cycles are homologically trivial, and while they depend on the choice of the base point when they are considered modulo rational equivalence, they are independent of this choice modulo algebraic equivalence.

The study of Ceresa cycles of Fermat curves has a rich history, going back to Bruno Harris’ fundamental work in the early 80s, where he showed via a Hodge-theoretic argument that the Ceresa cycle of the Fermat curve F(4) of degree 4 is algebraically nontrivial, thereby giving the first explicit example of an algebraic cycle that is homologically trivial but algebraically nontrivial. Soon after, Bloch used an l-adic argument to show that the Ceresa cycle of F(4) is, in fact, of infinite order modulo algebraic equivalence.

Since then, Harris’ and Bloch’s approaches have been adopted to other Fermat curves (in particular, by Otsubo, Tadokoro, and Kimura), giving rise to many interesting results. However, despite these efforts, until very recently the nontriviality of Ceresa cycles of Fermat curves modulo rational equivalence (let alone, algebraic equivalence) was only known unconditionally in very few cases.

The goal of this talk is to discuss some recent developments in the direction of this problem. The talk is based on a joint work with K. Murty.