$\mathbb{Q}$-curves, Hecke characters and some Diophantine equations

In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ for positive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$ is attached to each primitive solution, which happens to be a $\mathbb{Q}$-curve. Our main result is the construction of a Hecke character $\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$ extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures) corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and $\varepsilon$. Following some well known results and elimination techniques (together with some improvements) it provides a systematic procedure to study solutions of the above equations and allows us to prove non-existence of non-trivial primitive solutions for large values of $p$ of both equations for new values of $d$.