Lifting of Elliptic curves

Sanoli Gun and V. Kumar Murty

Vol. 301 (2019), No. 1, 101–106

DOI: 10.2140/pjm.2019.301.101

Abstract

Suppose that for all but finitely many primes $p$ we are given an elliptic curve $E_p$ defined over a finite field ${\mathbb{F}}_p$ of $p$ elements. We derive a criterion for there to exist an elliptic curve $E$ defined over $\mathbb{Q}$ for which the reduction of $E$ modulo $p$ is isogenous to $E_p$ for almost all $p$.

Keywords

lifting of elliptic curves, Chebotarev density theorem, generalized Riemann hypothesis, $\ell$-adic representations attached to elliptic curves

Mathematical Subject Classification 2010

Primary: 11F11, 11G05

Milestones

Received: 9 November 2018

Accepted: 6 April 2019

Published: 16 September 2019