Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant text] such the following holds. Let be a number field of degree let any rational prime is totally inert in and elliptic curve defined over has potentially multiplicative reduction at above text]. Then irreducible mod Galois representation. This result Diophantine applications within “modular method”. We present one application form Asymptotic version Fermat’s Last Theorem not been covered existing literature.