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Mathematics of Computation

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Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves


Author: Hao Chen
Journal: Math. Comp. 85 (2016), 2499-2514
MSC (2010): Primary 11G05
DOI: https://doi.org/10.1090/mcom3057
Published electronically: December 9, 2015
MathSciNet review: 3511290
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Abstract: Let $E$ be an optimal elliptic curve defined over $\mathbb {Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb {Q})$ generated by traces of branch points under a modular parametrization of $E$. We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of critical polynomials attached to $E$ and develop two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical polynomials. Finally, a table of critical polynomials is obtained for all elliptic curves of rank two and conductor smaller than 1000, from which we deduce our result.


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Additional Information

Hao Chen
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98115
Email: chenh123@uw.edu

Received by editor(s): January 17, 2015
Received by editor(s) in revised form: March 1, 2015, and March 18, 2015
Published electronically: December 9, 2015
Article copyright: © Copyright 2015 Hao Chen