On a special case of Watkins? conjecture

KAZALICKI, MATIJA; KOHEN, DANIEL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2017

0002-9939

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.