Solving Fermat-type equations over quadratic fields

Abstract

This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally real versus totally complex number fields. We use these techniques to study the generalized Fermat equation $d^ra^p+b^p+c^p=0$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ of class number one. Extending the results of Freitas\&Siksek and Turcas, we show that when $d=-3,-11,3,5,7,11,13,19,23$, there is an effective and explicit bound, depending on the field $\mathbb{Q}(\sqrt{d})$, such that the latter equation does not have certain types of special solutions. In addition, we obtain effective bounds for $d=-19,-43$, showing that the same equation has no non-trivial solutions of this type. We also discuss, for $d=6,14,21$, the solutions of a variant of the above equation. Our results over imaginary quadratic fields are conjectural. Serre's modularity conjecture and an analogue of Eichler-Shimura over totally complex fields are assumed.