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 | References | Similar Articles | Additional Information

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