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CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES

  • Fabrizio Barroero (a1)
  • Please note a correction has been issued for this article.

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 $\overline{\mathbb{Q}}$ .

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© Cambridge University Press 2017 

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

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References

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CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES

  • Fabrizio Barroero (a1)
  • Please note a correction has been issued for this article.

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