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

2014/12/09 by Chen, Hao · 1 citation
#11G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1412.2827

Abstract

Let E be an optimal elliptic curve defined over ℚ. The critical subgroup of E is defined by Mazur and Swinnerton-Dyer as the subgroup of E(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 describe 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.

Cited by

• Algorithms for Modular Parametrizations of Elliptic Curves over ℚ

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