Fonction zêta des hauteurs des surfaces de Hirzebruch dans le cas fonctionnel. (The height zeta-function of a Hirzebruch surface over a function field). (French) Zbl 1006.11032
Let \(k\) be a global field of positive characteristic, and let \(q\) stand for the cardinality of its field of constants. Given a positive integer \(m\), let \(V_m\) be the “Hirzebruch surface” defined by the equation \(y_1^mx_0= y_0^mx_1\) in the homogeneous coordinates \((x_0: x_1: x_2, y_0:y_1)\) on \(\mathbb{P}_k^2\times \mathbb{P}_k^1\), and let \(U\) be the open subset of \(V_m\) obtained by removing the line \(x_1= x_2= 0\). The author proves that the height zeta-function \(Z(s)= \sum_{x\in U(k)} H(x)^{-s}\), associated to the explicitly defined anti-canonical Arakelov height \(H\) on \(V_m\), is a rational function of \(q^{-s}\) having a second-order pole at \(s=1\), determines the residue as this pole, and points out that his results agree “with a version of Manin’s conjecture on the distribution of points of bounded height on almost-Fano varieties” over a function field \(k\) as above.
Reviewer: B.Z.Moroz (Bonn)
MSC:
| 11G50 | Heights |
| 14G05 | Rational points |
| 14G25 | Global ground fields in algebraic geometry |
| 14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
Keywords:
Manin’s conjecture on rational points; function fields; Hirzebruch surface; height zeta-functionReferences:
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