Unlikely Intersections in families of abelian varieties and the polynomial Pell equation

TL;DR

This paper investigates unlikely intersections in families of abelian varieties, proving finiteness results for points on curves not contained in special subgroup schemes, with applications to polynomial Pell equations.

Contribution

It extends previous results to powers of simple abelian schemes of dimension ≥2, completing the understanding of unlikely intersections in this context.

Findings

Finiteness of points on curves outside proper subgroup schemes

Extension of results to powers of simple abelian schemes

Applications to solvability of almost-Pell equations in polynomials

Abstract

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes. These results have applications in the study of solvability of almost-Pell equations in polynomials.