# Torsion points on families of simple abelian surfacesand Pell’s equation over polynomial rings

Masser, David and Zannier, Umberto. (2015) Torsion points on families of simple abelian surfacesand Pell’s equation over polynomial rings. Preprints Fachbereich Mathematik, 2015 (18).

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Official URL: https://edoc.unibas.ch/70012/

## Abstract

In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples by Bertrand. Furthermore there are applications to the study of Pell equations over polynomial rings; for example we deduce that there are at most finitely many complex t for which there exist $A, B \neq 0$ in $\mathbf{C}[X]$ with $A^2 - D B^2 = 1$ for $D = X^6 + X + t$. We also consider equations $A^2 - D B^2 = c^\prime X + c$, where the situation is quite different.
Faculties and Departments: 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Zahlentheorie (Masser)12 Special Collections > Preprints Fachbereich Mathematik Masser, David Preprint Universität Basel English 10.5451/unibas-ep70012 12 May 2019 21:18 28 Mar 2019 09:52

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