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CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES – CORRIGENDUM
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 3 / May 2020
- Published online by Cambridge University Press:
- 22 June 2018, p. 1029
- Print publication:
- May 2020
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- Article
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CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 5 / September 2019
- Published online by Cambridge University Press:
- 02 August 2017, pp. 941-956
- Print publication:
- September 2019
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- Article
- Export citation
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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 
$\overline{\mathbb{Q}}$.
