Let p be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ℚ, given by equations , 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 ≥ 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 ≤ 19.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8420-4-2016,
author = {Alain Kraus},
title = {Contre-exemples au principe de Hasse pour les courbes de Fermat},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {189-197},
zbl = {06602752},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8420-4-2016}
}
Alain Kraus. Contre-exemples au principe de Hasse pour les courbes de Fermat. Acta Arithmetica, Tome 172 (2016) pp. 189-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8420-4-2016/