On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one

Abstract: 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 $a,b,c \in K \setminus { 0 }$ such that $2 \mid abc$ and $p \geq 19$ is prime. The results are conjectural upon the veracity of a natural generalisation of Serre's modularity conjecture.