On the torsion locus of the Ceresa normal function

TL;DR

This paper proves that the positive-dimensional torsion locus of the Ceresa normal function in moduli space is not Zariski dense for genus g ≥ 3, with finitely many special components defined over algebraic numbers.

Contribution

It establishes non-density of the torsion locus of the Ceresa normal function and characterizes the special components with maximal Mumford-Tate groups.

Findings

Positive-dimensional torsion locus is not Zariski dense for g ≥ 3

Finitely many components have generic Mumford-Tate group equal to GSp_{2g}

Components are defined over algebraic numbers and stable under Galois action

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 $g\geq 3$. 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.