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Roots of unity and torsion points of abelian varieties

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Abstract

We answer a question raised by Hindry and Ratazzi concerning the intersection between cyclotomic extensions of a number field K and extensions of K generated by torsion points of an abelian variety over K. We prove that the property called \((\mu )\) in Hindry and Ratazzi (J Ramanujan Math Soc 25(1):81–111, 2010) holds for any abelian variety, while the same is not true for the stronger version of the property introduced in Hindry and Ratazzi (J Inst Math Jussieu 11(1):27–65, 2012).

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Acknowledgments

The author is grateful to Nicolas Ratazzi for attracting his interest to the problem considered in this paper. He also thanks Antonella Perucca for useful discussions and for pointing out Example 2, and the anonymous referee for suggesting that Theorem 1 could be made independent of the truth of the Mumford–Tate conjecture.

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  1. Département de Mathématiques d’Orsay, 91405, Orsay, France

    Davide Lombardo

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  1. Davide Lombardo
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Correspondence to Davide Lombardo.

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The author was supported by the Fondation Mathématique Jacques Hadamard, Grant “investissements d’avenir”.

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Lombardo, D. Roots of unity and torsion points of abelian varieties. Ramanujan J 43, 383–403 (2017). https://doi.org/10.1007/s11139-016-9813-1

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