In 2002 Watkins conjectured that given an elliptic curve defined over \({{\mathbb {Q}}}\) , 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 \({\mathbb {F}}_q(T)\) after extending constant scalars, and every quadratic twist of a modular elliptic curve over \({\mathbb {F}}_q(T)\) by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins’s conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins’s conjecture.
