Solving Fermat-type equations over quadratic fields
International Journal of Number Theory, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Publication Date: 2026
- Doi Number: 10.1142/s1793042126500922
- Journal Name: International Journal of Number Theory
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
- Keywords: elliptic curve, Galois representation, modular form, Modular method
- Open Archive Collection: AVESIS Open Access Collection
- Galatasaray University Affiliated: No
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 dr ap + bp + cp = 0 over quadratic fields ℚ(√d) of class number one. Extending the results of Freitas–Siksek and Turcas, we show that when d = −3, −11, −19, −43, 3, 5, 7, 11, 13, 19, 23, there is an effective and explicit bound, depending on the field ℚ(√d), such that the latter equation does not have certain types of special solutions. 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.