Abstract
Let \(K\) be a totally real field, and let \(S\) be a finite set of non-archimedean places of \(K\). It follows from the work of Merel, Momose and David that there is a constant \(B_{K,S}\) so that if \(E\) is an elliptic curve defined over \(K\), semistable outside \(S\), then for all \(p>B_{K,S}\), the representation \(\overline{\rho }_{E,p}\) is irreducible. We combine this with modularity and level lowering to show the existence of an effectively computable constant \(C_{K,S}\), and an effectively computable set of elliptic curves over \(K\) with CM \(E_1,\cdots ,E_n\) such that the following holds. If \(E\) is an elliptic curve over \(K\) semistable outside \(S\), and \(p>C_{K,S}\) is prime, then either \(\overline{\rho }_{E,p}\) is surjective, or \(\overline{\rho }_{E,p} \sim \overline{\rho }_{E_i,p}\) for some \(i=1,\dots ,n\).
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The authors are supported by EPSRC Programme Grant ‘LMF: L-Functions and Modular Forms’ EP/K034383/1. The second-named author is also supported by an EPSRC Leadership Fellowship EP/G007268/1.
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Anni, S., Siksek, S. On Serre’s uniformity conjecture for semistable elliptic curves over totally real fields. Math. Z. 281, 193–199 (2015). https://doi.org/10.1007/s00209-015-1478-8
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DOI: https://doi.org/10.1007/s00209-015-1478-8