Courbes de Fermat et principe de Hasse

TL;DR

This paper proves the existence of infinitely many Fermat curves over rational numbers with prime exponent p that are pairwise non-isomorphic and violate the Hasse principle.

Contribution

It establishes the infinite diversity of Fermat curves over Q of prime exponent p that do not satisfy the Hasse principle.

Findings

Existence of infinitely many non-isomorphic Fermat curves over Q

Counterexamples to the Hasse principle among these curves

Contradiction of the Hasse principle for infinitely many cases

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 shape $ax^p+by^p+cz^p=0$, 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.