Following the line of attack of La Bretèche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Châtelet surfaces which are defined as minimal proper smooth models of affine surfaces of the form Y² - aZ² = F(X,1), where a = -1, F ∈ ℤ[x₁,x₂] is a polynomial of degree 4 whose factorisation into irreducibles contains two non-proportional linear factors and a quadratic factor which is irreducible over ℚ [i]. This result deals with the last remaining case of Manin's conjecture for Châtelet surfaces with a = -1 and essentially settles Manin's conjecture for Châtelet surfaces with a < 0.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8312-2-2016,
author = {Kevin Destagnol},
title = {La conjecture de Manin pour certaines surfaces de Ch\^atelet},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {31-97},
zbl = {06602748},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8312-2-2016}
}
Kevin Destagnol. La conjecture de Manin pour certaines surfaces de Châtelet. Acta Arithmetica, Tome 172 (2016) pp. 31-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8312-2-2016/