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

TL;DR

This paper proves Manin's conjecture in its strong form for a specific family of Châtelet surfaces with certain polynomial conditions, completing the case for surfaces with negative parameter and advancing understanding of rational points distribution.

Contribution

It establishes the strong form of Manin's conjecture for a new class of Châtelet surfaces with particular polynomial factorizations, filling a key gap in the conjecture's verification.

Findings

Manin's conjecture proven for these Châtelet surfaces

Completes the case for surfaces with a=-1

Advances understanding of rational points on algebraic surfaces

Abstract

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 $a<0$.