Local criteria for the unit equation and the asymptotic Fermat's Last Theorem

TL;DR

This paper establishes local criteria in totally real number fields for the asymptotic Fermat's Last Theorem and the non-existence of solutions to the unit equation, based on ramification and splitting conditions.

Contribution

It introduces new local criteria involving ramification and splitting in totally real fields that ensure the asymptotic Fermat's Last Theorem holds.

Findings

Asymptotic FLT holds if 2 totally ramifies and 3 splits completely in F.

Provides local conditions for the non-existence of solutions to the unit equation.

Establishes criteria applicable to totally real fields of odd degree.

Abstract

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also for the non-existence of solutions to the unit equation over F. For example, if 2 totally ramifies and 3 splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.