Solving Fermat-type equations over quadratic fields

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 [Formula: see text] over quadratic fields [Formula: see text] of class number one. Extending the results of Freitas–Siksek and Turcas, we show that when [Formula: see text] −11, −19, −43, 3, 5, 7, 11, 13, 19, 23, there is an effective and explicit bound, depending on the field [Formula: see text], such that the latter equation does not have certain types of special solutions. We also discuss, for [Formula: see text], 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.