Q-curves, Hecke characters and some Diophantine equations II

Pacetti, Ariel ^[2] ; Villagra Torcomian, Lucas ^[1]
1. [1] Universidad Nacional de Córdoba
  
  Universidad Nacional de Córdoba
  
  Argentina
2. [2] Universidad de Aveiro. Department of Mathematics
Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 67, Nº 0, 2023, págs. 569-599
Idioma: inglés
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Resumen
- In the article [25] a general procedure to study solutions of the equations x4 − dy2 = 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(√d, √e)/Q(√d) (where e 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.