ℚ-curves, Hecke characters and some Diophantine equations II

• Autor(a) principal: Pacetti, Ariel
• Data de Publicação: 2023
• Outros Autores: Torcomian, Lucas Villagra
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10773/39463
• topic: Q-curves; Diophantine equations

In the article [25] 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 present article is to extend our previous results to positive values of d. On doing so, we give a description of the extension Q( \sqrt{d}, \sqrt{\varepsilon})/Q(\sqrt{d}) (where \varepsilon is a fundamental unit) needed to prove the existence of a Hecke character over Q(\sqrt{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.