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

TL;DR

This paper extends previous work on solutions to the equation x^4 - dy^2 = z^p to positive d, describing necessary field extensions and generalizing Galois representation results from imaginary to real quadratic fields.

Contribution

It broadens the analysis of Diophantine equations by including positive d and generalizes Galois image results for -curves from imaginary to real quadratic fields.

Findings

Describes the extension (d,(epsilon))/(d) for positive d.

Constructs Hecke characters with prescribed local conditions.

Generalizes large image Galois representation results to real quadratic fields.

Abstract

In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ 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 $\mathbb{Q}(\sqrt{d},\sqrt{\epsilon})/\mathbb{Q}(\sqrt{d})$ (where $\epsilon$ 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.