Watkins's conjecture for elliptic curves with a rational torsion

TL;DR

This paper investigates Watkins's conjecture relating the rank of elliptic curves over rationals to their modular degree, focusing on specific families with rational torsion and extending the inequality to broader contexts.

Contribution

It provides new insights into Watkins's conjecture for elliptic curves with rational torsion, especially within thin families and for various torsion structures.

Findings

Watkins's conjecture holds on average for certain elliptic curve families.

Quantitative results for elliptic curves with rational $ ext{ell}$-torsion.

Extension of the conjecture to inequalities involving additional parameters.

Abstract

Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $\nu_2 (m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that Watkins's conjecture holds on average. This article investigates the conjecture over certain thin families of elliptic curves. For example, for prime $\ell$, we quantify the elliptic curves featuring a rational $\ell$-torsion that satisfies Watkins's conjecture. Additionally, the study extends to a broader context, investigating the inequality $\mathrm{rank}(E(\mathbb{Q}))+M\leq \nu_2(m_E)$ for any positive integer $M$.