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 $M$.
Categories: math.NT