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

Abstract

Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over $\mathbf{Q}$ given by equations $a{x}^p+b{y}^p+c{z}^p=0$, 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 $p\ge 5$. 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 $p\le 19$.

arXiv commentin French