Ceresa cycles and the Northcott property

Let C+ be a curve of genus at least 2 embedded in its Jacobian and let C- = {-c : c in C+} be the negative embedding. The Ceresa cycle [C+] - [C-] is the simplest example of an algebraic cycle which is trivial in homology but (generally) non-trivial modulo algebraic equivalence. Hyperelliptic curves have trivial Ceresa class, but only recently examples of non-hyperelliptic curves with torsion Ceresa cycle were found. Gao—Zhang recently proved that the Beilinson—Bloch height of the Ceresa cycle satisfies a Northcott property on a certain mysterious open subset U_g of the moduli space M_g. In work with Jef Laga, we show how to find positive dimensional subvarieties of M_g not contained in U_g, for example the family of Picard curves y³ = x⁴ + ax² + bx + c in genus 3. The Northcott property fails on the Picard locus and the orders of the torsion Ceresa cycles are unbounded over the complex numbers. On the other hand, we find that there is a uniform bound on the order of a torsion (Picard curve) Ceresa cycle defined over a fixed number field.