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

Chen, Hao

doi:10.1090/mcom3057

Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves
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by Hao Chen;

Math. Comp. 85 (2016), 2499-2514

DOI: https://doi.org/10.1090/mcom3057

Published electronically: December 9, 2015

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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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