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Elliptic curves of odd modular degree

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Abstract

The modular degree m E of an elliptic curve E/Q is the minimal degree of any surjective morphism X 0(N) → E, where N is the conductor of E. We give a necessary set of criteria for m E to be odd. In the case when N is prime our results imply a conjecture of Mark Watkins. As a technical tool, we prove a certain multiplicity one result at the prime p = 2, which may be of independent interest.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, IL, 60208, USA

    Frank Calegari & Matthew Emerton

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  1. Frank Calegari
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  2. Matthew Emerton
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Correspondence to Frank Calegari.

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Supported in part by the American Institute of Mathematics.

Supported in part by NSF grant DMS-0401545.

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Calegari, F., Emerton, M. Elliptic curves of odd modular degree. Isr. J. Math. 169, 417–444 (2009). https://doi.org/10.1007/s11856-009-0017-x

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  • DOI: https://doi.org/10.1007/s11856-009-0017-x

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