Q-CURVES, HECKE CHARACTERS, AND SOME DIOPHANTINE EQUATIONS II

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, & RADIC; & RADIC; we give a description of the extension Q( d, & RADIC;�)/Q( d) (where E is a fundamental & RADIC; 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.