CMSA Algebra Seminar: 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 positive dimensional families of curves where only finitely many members of the family have torsion Ceresa cycle. The main idea is to study covers of curves with many automorphisms, and we will explain how we use the covering maps together with results on unlikely intersections in abelian varieties to construct such families. This is joint work with Tejasi Bhatnagar, Sheela Devadas and Toren D’Nelly Warady.