Given an elliptic curve $E/\mathbb{ Q}$ 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 $\mathrm{ deg}(\phi_E)$ and shed some light on Watkins's conjecture, which predicts $2^{ \mathrm{ rank}(E(\mathbb{ Q}))} \mid \mathrm{ deg}(\phi_E)$. 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.
Categories: math.NT