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arXiv · 1309.4748

Criteria for irreducibility of mod p representations of Frey curves

Nuno Freitas·Samir 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).

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Nuno Freitas, Samir Siksek. 2014-07-16. Criteria for irreducibility of mod p representations of Frey curves. https://arxiv.org/abs/1309.4748

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