Modular degree and a conjecture of Watkins

Abstract

Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a surjective morphism ${\varphi }_E:X_0(N)\to E$ defined over $\mathbb{Q}$. In this article, we discuss the growth of $\mathrm{deg}({\varphi }_E)$ and shed some light on Watkins's conjecture, which predicts $2^{\mathrm{rank}(E(\mathbb{Q}))}\mid \mathrm{deg}({\varphi }_E)$. Moreover, for any elliptic curve over ${\mathbb{F}}_q(T)$, we have an analogous modular parametrization relating to the Drinfeld modular curves. In this case, we also discuss growth and the divisibility properties.

21 pages, incorporated referee's suggestions