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On a special case of Watkins' conjecture


Authors: Matija Kazalicki and Daniel Kohen
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 11G05; Secondary 11G20
DOI: https://doi.org/10.1090/proc/13759
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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Additional Information

Matija Kazalicki
Affiliation: Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
Email: matija.kazalicki@math.hr

Daniel Kohen
Affiliation: Departamento de Matemática, Universidad de Buenos Aires and IMAS-CONICET, Ciudad Universitaria, Buenos Aires Argentina
Email: dkohen@dm.uba.ar

DOI: https://doi.org/10.1090/proc/13759
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