Criteria for irreducibility of mod p representations of Frey curves

2013/09/18 by Nuno Freitas, Samir Siksek, Freitas, Nuno +1 · 2 citations
Computer Science · Mathematics · #11F80 #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1309.4748

openalex publication_date 2013/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a totally real Galois number field and let A 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 not in P and E in A, the representation on 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 an existing result for Fermat-type equations of signature (13, 13, p).

Citations

• Caractère d'isogénie et critères d'irréductibilité
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• Effective Generalized Fermat equation of signature (2p, 2q, r) with odd narrow class number
• On Fermat's Last Theorem over the ℤ₃-extension of ℚ and other fields

Related

• Caractère d'isogénie et critères d'irréductibilité
• Critère d'irréductibilité pour les courbes elliptiques semi-stables sur un corps de nombres
• On the surjectivity of mod \ℓ representations associated to elliptic\n curves
• Masser-Wüstholz bound for reducibility of Galois representations for Drinfeld modules of arbitrary rank
• Galois representations attached to Q-curves and the generalized Fermat equation A⁴ + B² = C^p