Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat’s Last Theorem over Under the same assumption, we also prove that, for all prime exponents Fermat’s equation does not have non-trivial solutions over and

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] 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.

In the present article, we extend previous results of the author and we show that when K is any quadratic imaginary field of class number one, Fermat's equation a p + b p + c p = 0 does not have integral coprime solutions a, b, c ∈ K \ {0} such that 2 | abc and p ≥ 19 is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.

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