Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

Streszczenie

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 $a \lt 0$.