On Manin's conjecture for a family of Ch\^atelet surfaces

TL;DR

This paper proves Manin's conjecture for a specific family of Châtelet surfaces over Q, characterized by a particular polynomial form, despite these surfaces not satisfying weak approximation.

Contribution

The paper establishes the Manin conjecture for Châtelet surfaces defined by Y^2+Z^2=f(X) with a totally reducible cubic polynomial, expanding the class of surfaces where the conjecture is verified.

Findings

Manin's conjecture is confirmed for these Châtelet surfaces.

The surfaces do not satisfy weak approximation.

The result applies to surfaces with specific polynomial structures.

Abstract

The Manin conjecture is established for Ch\^atelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation.