Number Theory Seminar

Speaker: David Lilienfeldt (Hebrew University of Jerusalem)

Title: Experiments with Ceresa classes of cyclic Fermat quotients

Abstract: Let C be a curve of genus g > 2 embedded in its Jacobian J. The Ceresa cycle C-[-1]*C is a homologically trivial algebraic cycle of codimension g-1 on J. When C is hyperelliptic, this cycle is trivial modulo algebraic equivalence, whereas for a general curve C it is non-trivial by work of Ceresa. Recently, the first example of a non-hyperelliptic curve with torsion Ceresa cycle modulo algebraic equivalence was found by Beauville and Schoen. Inspired by their work, we give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images under the complex Abel-Jacobi map. All three examples (including the one of
Beauville and Schoen) are cyclic quotients of Fermat curves. In all three cases we compute the central order of vanishing of the L-function of the relevant motive. For our genus 3 example, the central L-value is non-vanishing and the cycle is torsion modulo algebraic equivalence, consistent with the conjectures of Beilinson and Bloch. This is joint work with Ari Shnidman.