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 Mg is not Zariski dense when g≥ 3. Moreover, it has only finitely many components with generic Mumford-Tate group equal to GSp2g; these components are defined over Q̄, and their union is closed under the action of Gal(Q̄/ Q). More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.