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).

Reçu le : 2013-10-05
Accepté le : 2014-01-13
Publié le : 2015-05-21
DOI : https://doi.org/10.5802/jtnb.894
Classification : 11F80,  11G05
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     author = {Nuno Freitas and Samir Siksek},
     title = {Criteria for Irreducibility of mod $p$ Representations of Frey Curves},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {1},
     year = {2015},
     pages = {67-76},
     doi = {10.5802/jtnb.894},
     mrnumber = {3346965},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2015__27_1_67_0/}
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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/item/JTNB_2015__27_1_67_0/

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