Vol. 308, No. 2, 2020

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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
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
Authors
Nathan Jones
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
Chicago, IL
United States