On Manin’s conjecture for a family of Châtelet surfaces

Régis de la Bretèche; Tim Browning; Emmanuel Peyre

doi:https://doi.org/10.4007/annals.2012.175.1.8

Pages 297-343 from Volume 175 (2012), Issue 1 by Régis de la Bretèche, Tim Browning, Emmanuel Peyre

Abstract

The Manin conjecture is established for Châtelet surfaces over $\mathbf{Q}$ arising as minimal proper smooth models of the surface $Y^2+Z^2=f(X)$ in $\mathbf{A}_{\mathbf{Q}}^3$, where $f\in \mathbf{Z}[X]$ is a totally reducible polynomial of degree $3$ without repeated roots. These surfaces do not satisfy weak approximation.

Milestones

Received: 2 February 2010
Revised: 5 November 2010
Accepted: 23 March 2011
Published online: 1 January 2012

Authors

Régis de la Bretèche

Institut de Mathématiques de Jussieu
UMR 7586 Case 7012
Université Paris 7 -- Denis Diderot
2, place Jussieu
F-75251 Paris cedex 05
France

Tim Browning

School of Mathematics
University of Bristol
Bristol BS8 1TW
England

Emmanuel Peyre

Institut Fourier
UMR 5582 du CNRS
Université Joseph Fourier
BP 74
38042 Saint-Martin d'Hères
France