On the asymptotic Fermat's Last Theorem over number fields

TL;DR

Under standard conjectures, the paper proves the asymptotic Fermat's Last Theorem for certain imaginary quadratic fields and provides a criterion involving S-unit equations for general number fields.

Contribution

It establishes the asymptotic Fermat's Last Theorem over specific imaginary quadratic fields and introduces a new criterion based on S-unit equations for general number fields.

Findings

Proves the theorem for imaginary quadratic fields with -d=2, 3 mod 4.

Provides a criterion for general number fields based on S-unit solutions.

Relies on standard conjectures from the Langlands Programme.

Abstract

Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.