Modular degree and a conjecture of Watkins

TL;DR

This paper investigates the growth of the degree of modular parametrizations of elliptic curves over $\,\mathbb{Q}$ and function fields, exploring Watkins's conjecture on divisibility related to the rank of the curve.

Contribution

It analyzes the growth of modular degrees and provides insights into Watkins's conjecture, extending the discussion to elliptic curves over function fields with Drinfeld modular curves.

Findings

Growth patterns of modular degrees for elliptic curves over $\,\mathbb{Q}$

Divisibility properties related to Watkins's conjecture

Analogous results for elliptic curves over function fields

Abstract

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.