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arXiv · 1904.10431

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

Nathan Jones
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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.

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Nathan Jones. 2019-04-23. A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve. https://doi.org/10.2140/pjm.2020.308.307

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