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La conjecture de Manin pour certaines surfaces de Châtelet

Volume 174 / 2016

Kevin Destagnol Acta Arithmetica 174 (2016), 31-97 MSC: 11D45, 11N37, 11D57. DOI: 10.4064/aa8312-2-2016 Published online: 10 June 2016

Abstract

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 $a \lt 0$.

Authors

  • Kevin DestagnolInstitut de Mathématiques de Jussieu-Paris Rive Gauche
    UMR 7586
    Université Paris Diderot-Paris 7
    Case postale 6052
    Bâtiment Sophie Germain
    75205 Paris Cedex 13, France
    URL: webusers.imj-prg.fr/~kevin.destagnol/
    e-mail

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