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This article is available for purchase or by subscription. See below.
Abstract
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Given an elliptic curve
without complex multiplication defined over a number field
, consider
the image of the Galois representation defined by letting Galois act on the torsion of
.
Serre’s open image theorem implies that there is a positive integer
for
which the Galois image is completely determined by its reduction modulo
. We prove a bound
on the smallest such
in terms of standard invariants associated with
. The
bound is sharp and improves upon previous results.
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Keywords
elliptic curves, Galois representations
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Mathematical Subject Classification 2010
Primary: 11F80, 11G05
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Milestones
Received: 18 October 2019
Revised: 8 May 2020
Accepted: 8 May 2020
Published: 9 December 2020
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