A conjecture of Watkins for quadratic twists

Jose A. Esparza-Lozano; Hector Pasten

A conjecture of Watkins for quadratic twists

Number Theory 2021-02-09 v1

Authors: Jose A. Esparza-Lozano , Hector Pasten

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.

Keywords

Watkins conjecture elliptic curve quadratic twist Mordell-Weil rank modular degree rational 2-torsion

Cite

@article{arxiv.2102.04185,
  title  = {A conjecture of Watkins for quadratic twists},
  author = {Jose A. Esparza-Lozano and Hector Pasten},
  journal= {arXiv preprint arXiv:2102.04185},
  year   = {2021}
}

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4 pages

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