La conjecture de Manin pour certaines surfaces de Châtelet

Kevin Destagnol¹

¹ Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR 7586 Université Paris Diderot-Paris 7 Case postale 6052 Bâtiment Sophie Germain 75205 Paris Cedex 13, France URL: <a href="http://webusers.imj-prg.fr/~kevin.destagnol/">webusers.imj-prg.fr/~kevin.destagnol/</a>

Acta Arithmetica, Tome 174 (2016) no. 1, pp. 31-97

Résumé

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 $$ Y^2-aZ^2=F(X,1), $$ where $a=-1$, $F \in \mathbb{Z}[x_1,x_2]$ is a polynomial of degree 4 whose factorisation into irreducibles contains two non-proportional linear factors and a quadratic factor which is irreducible over $\mathbb{Q}[i]$. 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 $a \lt 0$.

Détail

DOI : 10.4064/aa8312-2-2016

Mots-clés : following line attack bret che browning peyre prove manin conjecture its strong form conjectured peyre family telet surfaces which defined minimal proper smooth models affine surfaces form az where mathbb polynomial degree whose factorisation irreducibles contains non proportional linear factors quadratic factor which irreducible mathbb result deals remaining manin conjecture telet surfaces nbsp essentially settles manin conjecture telet surfaces