Consider the locus of the moduli space of genus 3 curves where the Ceresa cycle is torsion (modulo rational or algebraic equivalence). This locus is a countable union of proper closed algebraic subvarieties and contains the hyperelliptic locus, but little is known beyond this. I will report on joint work with Ari Shnidman, where we study this locus for curves with extra automorphisms, focusing on Picard curves.