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# Mathematical Research Letters

## Volume 26 (2019)

### Number 4

### Prime twists of elliptic curves

Pages: 1187 – 1195

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n4.a10

#### Authors

#### Abstract

For certain elliptic curves $E / \mathbb{Q}$ with $E(\mathbb{Q})[2] = \mathbb{Z} / 2 \mathbb{Z}$, we prove a criterion for prime twists of $E$ to have analytic rank $0$ or $1$, based on a $\mathrm{mod} \: 4$ congruence of $2$-adic logarithms of Heegner points. As an application, we prove new cases of Silverman’s conjecture that there exists a positive proposition of prime twists of $E$ of rank zero (resp. positive rank).

Received 27 November 2017

Accepted 24 May 2018

Published 25 October 2019