Criteria for Irreducibility of mod $p$ Representations of Frey Curves

Nuno Freitas; Samir Siksek

doi:10.5802/jtnb.894

Criteria for Irreducibility of mod

p

Representations of Frey Curves

Nuno Freitas ¹; Samir Siksek ²

¹ Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany
² Mathematics Institute University of Warwick CV4 7AL United Kingdom

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 67-76.

Abstract
Résumé

Let $K$ be a totally real Galois number field and let $ℰ$ 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 $p \notin 𝒫$ and $E \in ℰ$ , the representation $Gal (\bar{K} / K) \to Aut (E [p])$ 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 $(13, 13, p)$ .

Soit $K$ un corps de nombres galoisien totalement réel, et soit $ℰ$ 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 $p \notin 𝒫$ et $E \in ℰ$ , la représentation $Gal (\bar{K} / K) \to Aut (E [p])$ 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 $(13, 13, p)$ .

MR: 3346965

DOI: 10.5802/jtnb.894

Classification: 11F80, 11G05

Author's affiliations:

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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     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},
     pages = {67--76},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
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     mrnumber = {3346965},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.894/}
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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, Volume 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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