Unlikely intersections in the Torelli locus and the G-functions method

TL;DR

This paper proves finiteness results for special points on Hodge generic curves in the Torelli locus using height bounds derived from G-functions, extending André's method to odd weight variations.

Contribution

It extends André's G-functions method to odd weight variations and establishes Zilber-Pink-type finiteness results for Hodge generic curves intersecting the boundary of the Torelli locus.

Findings

Finiteness of points with non-simple Jacobians on certain curves

Height bounds for exceptional points in Hodge structures

Extension of G-functions method to odd weight cases

Abstract

Consider a smooth irreducible Hodge generic curve $S$ defined over $\bar{\Q}$ in the Torelli locus $T_g\subset \mathcal{A}_g$. We establish Zilber-Pink-type statements for such curves depending on their intersection with the boundary of the Baily-Borel compactification of $\mathcal{A}_g$. For example, when our curve intersects the $0$-dimensional stratum of this boundary and $g$ is odd, we show that there are only finitely many points in the curve for which the corresponding Jacobian variety is non-simple. These results follow as a special case of height bounds for exceptional points in $1$-parameter variations of geometric Hodge structures via Andr\'e's G-functions method, which we extend here to the setting of such variations of odd weight.