La conjecture de Manin pour certaines surfaces de Ch\^atelet

Following the line of attack from La Bret\`eche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Ch\^atelet 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 Ch\^atelet surfaces with $a=-1$ and essentially settles Manin's conjecture for Ch\^atelet surfaces with $a<0$.