A bound for the conductor of an open subgroup of $\mathrm{GL_2}$ associated to an elliptic curve

Nathan Jones

Vol. 308 (2020), No. 2, 307–331

DOI: 10.2140/pjm.2020.308.307

Abstract

Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre’s open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. We prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.

Keywords

elliptic curves, Galois representations

Mathematical Subject Classification 2010

Primary: 11F80, 11G05

Milestones

Received: 18 October 2019

Revised: 8 May 2020

Accepted: 8 May 2020

Published: 9 December 2020