On the torsion locus of the Ceresa normal function

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 ${\mathrm{G}\mathrm{S}\mathrm{p}}_{2g}$; these components are defined over $\bar{\mathbb{Q}}$, and their union is closed under the action of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.