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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A conjecture of Watkins for quadratic twists
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by Jose A. Esparza-Lozano and Hector Pasten PDF
Proc. Amer. Math. Soc. 149 (2021), 2381-2385 Request permission

Abstract:

Watkins conjectured that for an elliptic curve $E$ over $\mathbb {Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.
References
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Additional Information
  • Jose A. Esparza-Lozano
  • Affiliation: African Institute for Mathematical Sciences, Rue KG590 ST, Kigali, Rwanda
  • Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan
  • ORCID: 0000-0001-6555-2189
  • Email: josealanesparza@gmail.com
  • Hector Pasten
  • Affiliation: Pontificia Universidad Católica de Chile, Facultad de Matemáticas, 4860 Av. Vicuña Mackenna, Macul, RM, Chile
  • MR Author ID: 891758
  • Email: hector.pasten@mat.uc.cl
  • Received by editor(s): March 25, 2020
  • Received by editor(s) in revised form: October 13, 2020
  • Published electronically: March 25, 2021
  • Additional Notes: The first author was supported by a Carroll L. Wilson Award and the MIT International Science and Technology Initiatives (MISTI) during an academic visit to Pontificia Universidad Católica de Chile.
    The second author was supported by FONDECYT Regular grant 1190442.
  • Communicated by: Matthew Papanikolas
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 2381-2385
  • MSC (2020): Primary 11G05; Secondary 11F11, 11G18
  • DOI: https://doi.org/10.1090/proc/15376
  • MathSciNet review: 4246791