A conjecture of Watkins for quadratic twists

TL;DR

This paper proves Watkins' conjecture for quadratic twists of elliptic curves with non-trivial rational 2-torsion, showing the modular degree divisibility by powers of 2 related to the rank.

Contribution

It establishes Watkins' conjecture for a broad class of quadratic twists of elliptic curves with rational 2-torsion, expanding the cases where the conjecture is verified.

Findings

Watkins' conjecture holds for quadratic twists with many prime factors.

The modular degree is divisible by 2^r, where r is the Mordell-Weil rank.

The result applies to elliptic curves with non-trivial rational 2-torsion.

Abstract

Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. 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.