Criteria for irreducibility of mod p representations of Frey curves
Authors:
- Nuno FreitasNFNuno Freitas
- Samir SiksekSSSamir Siksek
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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).Keywords
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This publication has 5 references indexed in Scilit:
- Fermat-type equations of signature $(13,13,p)$ via Hilbert cuspformsMathematische Annalen, 2013
- COURBES ELLIPTIQUES SEMI-STABLES SUR LES CORPS DE NOMBRESInternational Journal of Number Theory, 2007
- Courbes elliptiques semi-stables et corps quadratiquesJournal of Number Theory, 1996
- Bornes pour la torsion des courbes elliptiques sur les corps de nombresInventiones Mathematicae, 1996
- Rational isogenies of prime degreeInventiones Mathematicae, 1978