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A bound for the conductor of an open subgroup of \(\mathrm{GL}_2\) associated to an elliptic curve. (English) Zbl 1465.11144

Summary: 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.

MSC:

11F80 Galois representations
11G05 Elliptic curves over global fields
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References:

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