On the torsion locus of the Ceresa normal function

Abstract

We prove that the positive-dimensional part of the torsion locus of the Ceresa normal function in ${\mathcal{M}}_g$ is not Zariski dense when $g\ge 3$. Moreover, it has only finitely many components with generic Mumford-Tate group equal to ${\mathrm{GSp}}_{2g}$; these components are defined over $\overline{\mathbb{Q}}$, and their union is closed under the action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.

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