Watkins's conjecture for elliptic curves with a rational torsion

Watkins's conjecture suggests that for an elliptic curve E/Q, the rank of the group E(Q) of rational points is bounded above by ν₂ (mE), where mE is the modular degree associated with E. It is known that Watkins's conjecture holds on average. This article investigates the conjecture over certain thin families of elliptic curves. For example, for prime , we quantify the elliptic curves featuring a rational -torsion that satisfies Watkins's conjecture. Additionally, the study extends to a broader context, investigating the inequality rank(E(Q))+M≤ ν₂(mE) for any positive integer M.