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On a special case of Watkins' conjecture

Authors: Matija Kazalicki and Daniel Kohen
Journal: Proc. Amer. Math. Soc. 146 (2018), 541-545
MSC (2010): Primary 11G05; Secondary 11G20
Published electronically: September 6, 2017
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Abstract: Watkins' conjecture asserts that for a rational elliptic curve $ E$ the degree of the modular parametrization is divisible by $ 2^r$, where $ r$ is the rank of $ E$. In this paper, we prove that if the modular degree is odd, then $ E$ has rank zero. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant.

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  • [1] Frank Calegari and Matthew Emerton, Elliptic curves of odd modular degree, Israel J. Math. 169 (2009), 417-444. MR 2460912,
  • [2] Henri Darmon, Rational points on modular elliptic curves, CBMS Regional Conference Series in Mathematics, vol. 101, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020572
  • [3] Benedict H. Gross, Heights and the special values of $ L$-series, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115-187. MR 894322
  • [4] Benedict H. Gross and Stephen S. Kudla, Heights and the central critical values of triple product $ L$-functions, Compositio Math. 81 (1992), no. 2, 143-209. MR 1145805
  • [5] Matija Kazalicki and Daniel Kohen, Supersingular zeros of divisor polynomials of elliptic curves of prime conductor, Res. Math. Sci. 4 (2017), Paper No. 10, 15. MR 3647576,
  • [6] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986) Nagoya Univ., Nagoya, 1986, pp. 217-242 (French). MR 891898
  • [7] William Stein and Mark Watkins, Modular parametrizations of Neumann-Setzer elliptic curves, Int. Math. Res. Not. 27 (2004), 1395-1405. MR 2052021,
  • [8] Soroosh Yazdani, Modular abelian varieties of odd modular degree, Algebra Number Theory 5 (2011), no. 1, 37-62. MR 2833784,
  • [9] Mark Watkins, Computing the modular degree of an elliptic curve, Experiment. Math. 11 (2002), no. 4, 487-502 (2003). MR 1969641

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Additional Information

Matija Kazalicki
Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Daniel Kohen
Affiliation: Departamento de Matemática, Universidad de Buenos Aires and IMAS-CONICET, Ciudad Universitaria, Buenos Aires Argentina

Received by editor(s): January 20, 2017
Received by editor(s) in revised form: March 31, 2017
Published electronically: September 6, 2017
Additional Notes: The first author’s work was supported by the QuantiXLie Center of Excellence
The second author’s work was supported by a doctoral fellowship of the Consejo Nacional de Inevsitagciones Científicas y Técnicas
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2017 American Mathematical Society

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