We will talk about the Ceresa class, which is the image under a cycle class map of a canonical homologically trivial algebraic cycle associated to a curve in its Jacobian. In his 1983 thesis, Ceresa showed that the generic curve of genus at least 3 has nonvanishing Ceresa cycle modulo algebraic equivalence. Strategies for proving Fermat curves have infinite order Ceresa cycles due to B. Harris, Bloch, Bertolini-Darmon-Prasanna, Eskandari-Murty use a variety of ideas ranging from computation of explicit iterated period integrals, special values of p-adic L functions and points of infinite order on the Jacobian of Fermat curves. In fact, Bloch's results about the Ceresa cycle of Fermat quartics provided the first concrete evidence for the generalization of the BSD conjecture to the Bloch-Beilinson conjectures.

We will survey several recent results about the Ceresa cycle and the Ceresa class. The Ceresa class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. We will present joint work with Dean Bisogno, Wanlin Li and Daniel Litt, where we construct a non-hyperelliptic genus 3 quotient of the Fricke--Macbeath curve with torsion Ceresa class, using the character theory of the automorphism group of the curve, namely, PSL2(F8).