Relative Manin-Mumford for semi-abelian surfaces

TL;DR

This paper proves that Ribet sections are the main obstacle to the relative Manin-Mumford conjecture for semi-abelian surfaces, with implications for Zilber-Pink conjecture cases and polynomial Pell equations.

Contribution

It identifies Ribet sections as the sole obstruction to the relative Manin-Mumford conjecture in this setting and explores related applications.

Findings

Ribet sections are the only obstructions for the conjecture.

Applications to Zilber-Pink conjecture for certain Shimura varieties.

Insights into polynomial Pell equations with non-separable discriminants.

Abstract

We show that Ribet sections are the only obstruction to the validity of the relative Manin-Mumford conjecture for one dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber-Pink conjecture for curves in a mixed Shimura variety of dimension four, as well as the study of polynomial Pell equations with non-separable discriminants.