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Nuno Freitas; Samir Siksek
Criteria for Irreducibility of mod $p$ Representations of Frey Curves
Journal de théorie des nombres de Bordeaux, 27 no. 1 (2015), p. 67-76, doi: 10.5802/jtnb.894
Article PDF | Reviews MR 3346965
Class. Math.: 11F80, 11G05

Résumé - Abstract

Let $K$ be a totally real Galois number field and let $\mathcal{E}$ be a set of elliptic curves over $K$. We give sufficient conditions for the existence of a finite computable set of rational primes $\mathcal{P}$ such that for $p \notin \mathcal{P}$ and $E \in \mathcal{E}$, the representation $\operatorname{Gal}(\overline{K}/K) \rightarrow \operatorname{Aut}(E[p])$ is irreducible. Our sufficient conditions are often satisfied for Frey elliptic curves associated to solutions of Diophantine equations; in that context, the irreducibility of the mod $p$ representation is a hypothesis needed for applying level-lowering theorems. We illustrate our approach by improving on a result of [6] for Fermat-type equations of signature $(13,13,p)$.

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