Fermat's Last Theorem and modular curves over real quadratic fields

Philippe Michaud-Jacobs
2022 Acta Arithmetica  
There has been much recent interest in the study of the Fermat equation x n + y n = z n over number fields. Following in the footsteps of Wiles [41], we would ideally like to show that this equation has no non-trivial solutions for n ≥ 4 and x, y, z ∈ K, a number field. By a non-trivial solution, we mean xyz ̸ = 0. The study of the Fermat equation over number fields dates back to the work of Maillet in the late 19th century [13, p. 578]. Asymptotic versions of Fermat's Last Theorem over number
more » ... ields (proving there are no non-trivial solutions if all the prime factors of n are greater than some bound depending on K) have been proven over various number fields (see [16, 14] for example). In this paper, we are concerned with trying to prove the non-existence of non-trivial solutions for all n ≥ 4 over a real quadratic field K. Jarvis and Meekin [22] proved this statement over the field Q( √ 2). This work was then extended by Freitas and Siksek [18] to the quadratic fields Q( √ d), with d squarefree, d ̸ = 5, 17 in the range 3 ≤ d ≤ 23, where it was shown that there are no non-trivial solutions for n ≥ 4. Kraus [29] proved that for K a cubic field of discriminant 148, 404, or 564 there are no non-trivial solutions for n ≥ 4. The aim of this paper is to extend the result of Freitas and Siksek to squarefree d in the range 26 ≤ d ≤ 97, as well as introduce techniques that can be used to study other Diophantine equations, both over the rationals and number fields of low degree. For most values of d in this range, issues arise surrounding irreducibility
doi:10.4064/aa210812-2-4 fatcat:moh33whs2fdv5hpjmzpoi4j4ca