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

In this note we prove that for every integer $d\ge 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q>\max \{d-1,5\}$ be any rational prime that is totally inert in $K$ and $E$ any elliptic curve defined over $K$ such that $E$ has potentially multiplicative reduction at the prime $\mathfrak{q}$ above $q$. Then for every rational prime $p>B_d$, $E$ has an irreducible mod $p$ Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.