Contre-exemples au principe de Hasse pour les courbes de Fermat

TL;DR

This paper investigates Fermat curves over rationals and provides partial results supporting the conjecture that infinitely many such curves violate the Hasse principle, using cyclotomic methods and Chebotarev's density theorem.

Contribution

It proves the conjecture for primes up to 19 using cyclotomic techniques, advancing understanding of counterexamples to the Hasse principle for Fermat curves.

Findings

Proved the conjecture for p ≤ 19.

Used cyclotomic approach and Chebotarev's density theorem.

Partial progress towards the infinite counterexamples conjecture.

Abstract

Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ${\bf Q}$ given by equations $ax^p+by^p+cz^p=0$, with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent $p$ which are counterexamples to the Hasse principle. It is a consequence of the abc-conjecture if $p\geq 5$. Using a cyclotomic approach due to H. Cohen and Chebotarev's density theorem, we obtain a partial result towards this conjecture, by proving it for $p\leq 19$.