Let E E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487–502 (2003)] conjectured that the rank of E E is bounded by the 2 2 -adic valuation of the modular degree of E E . We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2 2 , provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.

Watkins conjectured that given an elliptic curve defined over Q, its Mordell-Weil rank is at most the 2-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over Fq(T ) after extending constant scalars, and every quadratic twist of a modular elliptic curve over Fq(T ) by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins' conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins' conjecture.

Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the 2-adic valuation of its modular degree. We show that this conjecture is satisfied when E is any quadratic twist of an elliptic curve with rational 2-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form y 2 = x 3 − dx, with d a biquadratefree integer.

For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$ , it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math.342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr.49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2)10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.