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According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations. However, given a galois representation, it is usually difficult ...

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. Serre's open image theorem (which appears in his book 'Abelian $l$-Adic Representations and Elliptic Curves'...