Covers of curves, Ceresa cycles, and unlikely intersections

The Ceresa cycle is a canonical homologically trivial algebraic cycle associated to a curve in its Jacobian. In his 1983 thesis, Ceresa showed that this cycle is algebraically nontrivial for a very general complex curve of genus at least 3. In the last few years, there have been many new results shedding light on the locus in the moduli space of genus g curves where the Ceresa cycle becomes torsion. We will survey these recent results and provide new examples of families of curves where only finitely many members of the family have torsion Ceresa cycle. The main idea is to leverage the covering map to reduce the question of torsionness of the Ceresa cycle to the torsionness of a canonical point on the Jacobian and combine this with results on unlikely intersections in abelian varieties. This is joint work with Tejasi Bhatnagar, Sheela Devadas and Toren D’Nelly Warady.