Le th\'eor\`eme de Fermat sur certains corps de nombres totalement r\'eels

TL;DR

This paper investigates Fermat's Last Theorem over totally real number fields, establishing criteria related to Hilbert modular forms and proving the theorem effectively for certain degrees using modular methods.

Contribution

It provides new criteria for Fermat's Last Theorem over totally real fields and applies modular methods to prove the theorem effectively for specific degrees.

Findings

Criteria for Fermat's Last Theorem over totally real fields based on Hilbert modular forms

Effective proof of Fermat's Last Theorem for degrees 3,4,5,6,8 over certain fields

Numerical tests are feasible when the narrow class number is 1

Abstract

Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat's Last Theorem is true over $K$, criterion related to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$. Furthermore, using the modular method, we prove Fermat's Last Theorem effectively, over some number fields whose degrees over $\mathbb{Q}$ are $3,4,5,6$ and $8$.