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arXiv · 1509.07060

La conjecture de Manin pour certaines surfaces de Châtelet

Kevin Destagnol
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Abstract

Following the line of attack from 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 $a<0$.

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Kevin Destagnol. 2016-02-15. La conjecture de Manin pour certaines surfaces de Châtelet. https://doi.org/10.4064/aa8312-2-2016

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