On Ternary Diophantine Equations of Signature $(p,p,3)$ over Number Fields

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. Firstly, by assuming some standard modularity conjecture we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Secondly, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\Q(\sqrt{-d})$ where $d=1,7,19,43,67$ whenever $p$ is bigger than this bound. During the course of the proof we prove various results about the irreducibility of Galois representations, image of inertia groups and Bianchi newforms.