Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins' conjecture

TL;DR

This paper investigates Watkins' conjecture relating the rank of elliptic curves to their modular degree, providing new cases where the conjecture holds, especially for curves with specific reduction properties and rational points of order two.

Contribution

The paper establishes new instances of Watkins' conjecture for elliptic curves with particular reduction types and torsion points, expanding the known cases where the conjecture is verified.

Findings

Watkins' conjecture holds for elliptic curves with additive reduction at 2.

The conjecture is verified for curves with good reduction outside at most two odd primes.

Elliptic curves with a rational point of order two satisfy Watkins' conjecture.

Abstract

Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the Mordell--Weil Theorem, $\mathsf{E}(\mathbb{Q})\simeq \mathbb{Z}^r \oplus T$ for some nonnegative integer $r$ and some finite group $T$. Watkins' Conjecture predicts that $2^r$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins' Conjecture, specifically for elliptic curves with additive reduction at $2$, good reduction outside of at most two odd primes, and a rational point of order two.