Le théorème de Fermat sur certains corps de nombres totalement réels

Abstract

Let $K$ be a totally real number field. For all prime number $p\ge 5$, 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 $8$.

in French

References in corpus (2)

• Fermat's Last Theorem over some small real quadratic fields
• Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell }+y^{2m}=z^p$

Cited by in corpus (1)

• Fermat's Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$