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

2019/09/02 by F. Barroero, L. Capuano · 4 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Polynomial and algebraic computation

paper · pdf · doi:10.1112/plms.12289

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 fibred 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 ⩾ 2 . This, combined with the above-mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes which has applications in the study of solvability of almost-Pell equations in polynomials.

Citations

• Courbes sur une vari�t� ab�lienne et points de torsion
• Torsion points on families of products of elliptic curves

Cited by

• COLLISION OF ORBITS ON AN ELLIPTIC SURFACE
• Isogeny relations in products of families of elliptic curves
• Singular intersections in families of abelian varieties
• Unlikely intersections in Shimura varieties and beyond: a survey

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