Courbes de Fermat et principe de Hasse
Volume 222 / 2026
Acta Arithmetica 222 (2026), 27-36
MSC: Primary 11D41
DOI: 10.4064/aa250111-4-8
Published online: 17 November 2025
Abstract
Let $p\geq 3$ be a prime number. A Fermat curve over $\mathbb {Q}$ of exponent $p$ is defined by an equation of the form $ax^p+by^p+cz^p=0$, where $a$, $b$, $c$ are non-zero rational numbers. We prove in this article that there exist infinitely many Fermat curves defined over $\mathbb {Q} $, of exponent $p$, pairwise non $\mathbb {Q}$-isomorphic, contradicting the Hasse principle.
