Class field theory, Diophantine analysis and the asymptotic Fermat's\n Last Theorem

2019/02/20 by Nuno Freitas, Freitas, Nuno, Alain Kraus +3 · 2 citations
Mathematics · #11D41 (Primary) #11J86 (Secondary) #11R37 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1902.07798

openalex publication_date 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria\nfor the asymptotic Fermat's Last Theorem to hold over a specific number field.\nThose works in turn build on many deep theorems in arithmetic geometry. In this\npaper we combine the aforementioned results with techniques from class field\ntheory, the theory of p-groups and p-extensions, Diophantine approximation and\nlinear forms in logarithms, to establish the asymptotic Fermat's Last Theorem\nfor many infinite families of number fields, and for thousands of number fields\nof small degree. For example, we prove the effective asymptotic Fermat's Last\nTheorem for the infinite family of fields \ℚ(\ζ_2^r)⁺.\n

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• Formally certifying number field invariants
• Non-trivial Solutions of Aa^p+Bb^p=Cc³ over Number Fields

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