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 M g $\mathcal {M}_g$ script upper M Subscript g is not Zariski dense when g ≥ 3 $g\geq 3$ g greater than or equals 3 . Moreover, it has only finitely many components with generic Mumford-Tate group equal to GSp 2 g $\mathrm {GSp}_{2g}$ upper G upper S p Subscript 2 g ; these components are defined over Q ― $\overline {\mathbb Q}$ double struck upper Q overbar , and their union is closed under the action of Gal ( Q ― / Q ) $\mathrm {Gal}(\overline {\mathbb Q}/\mathbb Q)$ upper G a l left parenthesis double struck upper Q overbar divided by double struck upper Q right parenthesis . More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.