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

Alain Kraus¹

¹ Université de Paris VI Institut de Mathématiques de Jussieu 4 Place Jussieu 75005 Paris, France

Acta Arithmetica, Tome 174 (2016) no. 2, pp. 189-197.

Résumé

Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over $\mathbb Q$, 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. This 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 $p\leq 19$.

Détail

DOI : 10.4064/aa8420-4-2016

Mots-clés : odd prime number paper concerned behaviour fermat curves defined mathbb given equations respect local global hasse principle conjectured there exist infinitely many fermat curves exponent which counterexamples hasse principle consequence abc conjecture geq using cyclotomic approach due cohen chebotarev density theorem obtain partial result towards conjecture proving leq