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

TL;DR

This paper investigates the structure of the critical subgroup of elliptic curves over rationals, proving it is torsion for all rank two curves with conductor under 1000, using new algorithms and polynomial analysis.

Contribution

Introduces critical polynomials and algorithms to determine when the critical subgroup is torsion, providing computational evidence for rank two elliptic curves under 1000.

Findings

Critical subgroup is torsion for all rank two curves with conductor < 1000

Developed algorithms to compute critical polynomials

Provided a comprehensive table of critical polynomials for these curves

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 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.