La conjecture de Manin pour certaines surfaces de Châtelet

Abstract

Following the line of attack from 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 $$Y^2-a{Z}^2=F(X,1),$$ 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 $a<0$.

54 pages, in French, accepted for publication in Acta Arithmetica