Criteria for Irreducibility of mod p Representations of Frey Curves
Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 67-76.

Soit K un corps de nombres galoisien totalement réel, et soit un ensemble de courbes elliptiques sur K. Nous donnons des conditions suffisantes pour l’existence d’un ensemble calculable de nombres premiers 𝒫 tels que, pour p𝒫 et E, la représentation Gal(K¯/K)Aut(E[p]) soit irréductible. Nos conditions sont en général satisfaites par les courbes de Frey associées à des solutions d’équations diophantiennes. Dans ce contexte, l’irréductibilité de la représentation mod p est une hypothèse requise pour l’application des théorèmes d’abaissement du niveau. Comme illustration de notre approche, nous avons amélioré le résultat de [6] pour les équations de Fermat de signature (13,13,p).

Let K be a totally real Galois number field and let be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes 𝒫 such that for p𝒫 and E, the representation Gal(K¯/K)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).

DOI : 10.5802/jtnb.894
Classification : 11F80, 11G05

Nuno Freitas 1 ; Samir Siksek 2

1 Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany
2 Mathematics Institute University of Warwick CV4 7AL United Kingdom
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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, Tome 27 (2015) no. 1, pp. 67-76. doi : 10.5802/jtnb.894. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.894/

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