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Elliptic curves with abelian division fields

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Abstract

Let E be an elliptic curve over \(\mathbb {Q}\), and let \(n\ge 1\). The central object of study of this article is the division field \(\mathbb {Q}(E[n])\) that results by adjoining to \(\mathbb {Q}\) the coordinates of all n-torsion points on \(E(\overline{\mathbb {Q}})\). In particular, we classify all curves \(E/\mathbb {Q}\) such that \(\mathbb {Q}(E[n])\) is as small as possible, that is, when \(\mathbb {Q}(E[n])=\mathbb {Q}(\zeta _n)\), and we prove that this is only possible for \(n=2,3,4\), or 5. More generally, we classify all curves such that \(\mathbb {Q}(E[n])\) is contained in a cyclotomic extension of \(\mathbb {Q}\) or, equivalently (by the Kronecker–Weber theorem), when \(\mathbb {Q}(E[n])/\mathbb {Q}\) is an abelian extension. In particular, we prove that this only happens for \(n=2,3,4,5,6\), or 8, and we classify the possible Galois groups that occur for each value of n.

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Acknowledgments

This project began after a conversation with J. M. Tornero about torsion points over cyclotomic fields, so we would like to thank him for the initial questions that inspired this work. We would also like to thank David Zywina for providing us with several models of modular curves. The second author would like to thank the Universidad Autónoma de Madrid, where much of this work was completed during a sabbatical visit, for its hospitality. The authors would also like to thank the editors and the referee for their comments and suggestions.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

    Enrique González–Jiménez

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Álvaro Lozano-Robledo

Authors
  1. Enrique González–Jiménez
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  2. Álvaro Lozano-Robledo
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Corresponding author

Correspondence to Álvaro Lozano-Robledo.

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The first author was partially supported by the Grant MTM2012–35849.

Appendix: Tables

Appendix: Tables

In this section we include four tables that summarize our findings and provide concrete examples of elliptic curves (or families of elliptic curves) with each possible isomorphism type of abelian division field, and torsion structure over \(\mathbb {Q}\) (Tables 1, 2, 3, 4).

Table 1 CM curves
Full size table
Table 2 \(n=3,5\), or 6
Full size table
Table 3 \(n=2\) or 4
Full size table
Table 4 \(n=8\)
Full size table

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González–Jiménez, E., Lozano-Robledo, Á. Elliptic curves with abelian division fields. Math. Z. 283, 835–859 (2016). https://doi.org/10.1007/s00209-016-1623-z

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  • DOI: https://doi.org/10.1007/s00209-016-1623-z

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