Number/Representation Theory

Ceresa cycles were introduced by Ceresa in the 1980s as one of the first examples of cycles able to distinguish between homological and algebraic equivalence. Recently, there has been renewed interest in the study of the vanishing of Ceresa cycles, also due to their connection to Gross–Kudla–Schoen diagonal cycles, which are used to study the positivity of the Beilinson–Bloch height, as well as conjectures involving $L$-functions.

In this talk, we introduce two generalizations of GKS-cycles and prove they “measure how much" Ceresa cycles vanish, extending work of S. Zhang and Moonen–Yin. We moreover prove an integral refinement of S. Zhang's result, relating the order of torsion of the Ceresa cycle with the one of the GKS-cycle. We then apply our general theory to prove (non-)vanishing results for generic curves, and if time permits bound the order of torsion in specific examples.

This is joint work with L. Lagarde, M. Moakher, J. Rawson, and F. Trejos-Suárez.