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

In the article [PV] a general procedure to study solutions of the equations $x^4-d{y}^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrt{ϵ})\mathrm{/}\mathbb{Q}(\sqrt{d})$ (where $ϵ$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some “large image” results due to Ellenberg regarding images of Galois representations coming from $\mathbb{Q}$-curves from imaginary to real quadratic fields.