Modular degree and a conjecture of Watkins

Given an elliptic curve $E\mathrm{/}\mathbb{Q}$ of conductor $N$, there exists a surjective morphism ${\phi }_E:X_0(N)\to E$ defined over $\mathbb{Q}$. In this article, we discuss the growth of $\mathrm{d}\mathrm{e}\mathrm{g}({\phi }_E)$ and shed some light on Watkins’s conjecture, which predicts $2^{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(E(\mathbb{Q}))}\mid \mathrm{d}\mathrm{e}\mathrm{g}({\phi }_E)$. 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.