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CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES

Published online by Cambridge University Press:  02 August 2017

Fabrizio Barroero*
Affiliation:
Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland (fbarroero@gmail.com)

Abstract

Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

Type
Research Article
Copyright
© Cambridge University Press 2017 

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Footnotes

F. B. was supported by the EPSRC grant EP/N007956/1’ and the SNF grant 165525.

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