On a special case of Watkins' conjecture

TL;DR

This paper proves that rational elliptic curves with odd modular degree have rank zero and confirms Watkins' conjecture for all rank two curves with prime conductor and positive discriminant.

Contribution

It establishes a new link between the parity of the modular degree and the rank of elliptic curves, and verifies Watkins' conjecture in specific cases.

Findings

Elliptic curves with odd modular degree have rank zero.

Watkins' conjecture holds for rank two curves with prime conductor and positive discriminant.

Abstract

Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree of the modular parametrization is divisible by $2^r$, where $r$ is the rank of $E$. In this paper we prove that if the modular degree is odd then $E$ has rank $0$. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant.