Q-curves, Hecke characters and some Diophantine equations II

Abstract

In the article [25] a general procedure to study solutions of the equations x4 − dy2 = zp was presented for negative values of d. The purpose of the present article is to extend our previous results to positive values of d. On doing so, we give a description of the extension Q(√d, √ε)/Q(√d) (where ε is a fundamental unit) needed to prove the existence of a Hecke character over Q(√d) with prescribed local conditions. We also extend some “large image” results due to Ellenberg regarding images of Galois representations coming from Q-curves from imaginary to real quadratic fields.