Mathematical Research Letters

Volume 26 (2019)

Number 4

Prime twists of elliptic curves

Pages: 1187 – 1195



Daniel Kriz (Department of Mathematics, Massachusetts Institute of Technology. Cambridge, Mass., U.S.A.)

Chao Li (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)


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