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Watkins conjectured that for an elliptic curve E over Q of Mordell-Weil rank r, the modular degree of E is divisible by 2. If E has non-trivial rational 2-torsion, we prove the conjecture for all the quadratic twists of E by squarefree integers with sufficiently many prime factors. 1. Ranks and modular degree For an elliptic curve E over Q of conductor N , the modularity theorem [25, 23, 4] gives a nonconstant morphism φE : X0(N) → E defined over Q where X0(N) is the modular curve associated to the congruence subgroup Γ0(N) ⊆ SL2(Z). We assume that φE has minimal degree and that it maps the cusp i∞ to the neutral point of E. These requirements uniquely determine φE up to sign. The modular degree of E is mE = degφE and it has profound arithmetic relevance; for instance, polynomial bounds for its size in terms of N are essentially equivalent to the abc conjecture [11, 17]. The 2-adic valuation is denoted by v2. Motivated by numerical data, Watkins [24] conjectured that v2(mE) for an elliptic curve E is closely related to the Mordell-Weil rank of E over Q. Conjecture 1.1 (Watkins). For every elliptic curve E over Q we have rankE(Q) ≤ v2(mE). Dummigan [8] showed that part of the conjecture would follow from strong R = T conjectures. Also, large part of Watkins’ conjecture is proved for elliptic curves of odd modular degree [5, 26, 12, 13], although it is not known whether there exist infinitely many elliptic curves of this kind [21]. The goal of this note is to prove Watkins’ conjecture unconditionally in several new cases. Let us introduce some notation. For an elliptic curve E and a fundamental (quadratic) discriminant D, the quadratic twist of E by D is denoted by E(D). The Manin constant of E is denoted by cE (cf. Section 2.3). The number of distinct prime factors of an integer n is ω(n). Theorem 1.2. Let E be an elliptic curve over Q of conductor N with non-trivial rational 2torsion. Assume that E has minimal conductor among its quadratic twists. If D is a fundamental discriminant with ω(D) ≥ 6 + 5ω(N)− v2(mE/c 2 E), then Watkins’ conjecture holds for E (D). The quantity 6+ 5ω(N)− v2(mE/c 2 E) is effectively computable and it can be read from existing tables of elliptic curves when N is not too large, see for instance [14]. For a positive integer A, it is a standard result of analytic number theory that the number of positive integers n up to x having ω(n) ≤ A is O(x(log log x)A−1/ log x). We deduce: Corollary 1.3. Let E be an elliptic curve over Q with non-trivial rational 2-torsion. There is an effective constant κ(E) depending only on E such that the number of fundamental discriminants D with |D| ≤ x such that Watkins’ conjecture fails for E(D) is bounded by O ( x(log log x)κ(E)/ log x )

A conjecture of Watkins for quadratic twists