A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve

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. In this note, 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.

A Bound for the Image Conductor of a Principally Polarized Abelian Variety with Open Galois Image

“…Because Theorem 1.2 is known [2] for g = 1, in order to simplify our exposition, g will always denote an integer that is at least two, unless otherwise stated. We shall often use the abbreviation ℓ g , which denotes…”

“…A proof of Proposition 4.1, for the case of g = 1, is given in [2,Proposition 1.4]. The proof readily generalizes, mutatis mutandis, to prove Proposition 4.1 for arbitrary g. For this reason, in this section we shall only say a few words to highlight the steps of the proof and refer the reader to [2] for the details.…”

“…In a recent paper [2], Jones established an upper bound for m A , in terms of standard invariants of A, in the case that A is an elliptic curve without complex multiplication. Further, he remarked that his techniques should be able to be extended to prove an analogous result for principally polarized abelian varieties of arbitrary dimension.…”