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

Freitas, Nuno¹  ; Siksek, Samir²

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, Tome 27 (2015) no. 1, pp. 67-76.

Abstract (VO)

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 $Gal(\overline{K}/K)\to Aut\left(E\left[p\right]\right)$ 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 $(13,13,p)$.

Résumé

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 $Gal(\overline{K}/K)\to Aut\left(E\left[p\right]\right)$ 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 $(13,13,p)$.

Détail

DOI : 10.5802/jtnb.894

Classification : 11F80, 11G05

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