Modular degree and a conjecture of Watkins

TL;DR

This work investigates the modular degree m_E of elliptic curves E/ℚ with conductor N, focusing on its growth, 2-adic valuation, and divisibility patterns predicted by Watkins's conjecture 2^{rank(E(ℚ))} | m_E. It develops and applies tools relating m_E to Manin’s constant, Petersson norms, and symmetric-square L-functions, and connects m_E to congruence numbers r_E via Ribet–ARS-type results, including Gross’s supersingular framework for prime conductors. The paper also analyzes special cases (prime and minimal conductors, curves with 2-torsion structures, and quadratic twists) and proposes conjectures on supersingular coefficients, establishing conditional and unconditional results toward Watkins’s conjecture, both over ℚ and in the function-field setting. Finally, it surveys growth bounds for m_E, highlights connections to L-functions and conjectures like ABC, BSD, and GRH, and extends the discussion to Drinfeld modular curves and analogous phenomena in the function-field context, offering a broad, multifaceted view of modular degrees and their arithmetic significance.

Abstract

Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, there exists a surjective morphism $φ_E: X_0(N) \to E$ defined over $\mathbb{Q}$. In this article, we discuss the growth of $\mathrm{deg}(φ_E)$ and shed some light on Watkins's conjecture, which predicts $2^{\mathrm{rank}(E(\mathbb{Q}))} \mid \mathrm{deg}(φ_E)$. 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.

Theorems & Definitions (36)

• Theorem 1.1
• Conjecture 1: Watkins
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• proof
• Theorem 3.1
• Lemma 3.2