Watkins’ modular-degree divisibility conjecture

Prove that for every elliptic curve E over Q, the modular degree m_E is divisible by 2 raised to the Mordell–Weil rank of E(Q), namely 2^{\operatorname{rank}E(Q)}\mid m_E.

Background

Watkins’ conjecture concerns a proposed lower bound on the 2-adic divisibility of the modular degree of an elliptic curve in terms of its Mordell–Weil rank. The paper establishes the divisibility 4\mid m_E for positive-rank elliptic curves of prime conductor with root number +1, and consequently verifies the conjecture for curves in this class having rank 2.

The general statement quoted here is broader than the cases proved in the paper: it applies to arbitrary elliptic curves over Q and arbitrary Mordell–Weil rank. Thus the paper’s results constitute partial verification rather than a proof of the full conjecture.