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Height zeta functions of fibre bundles over generalized flag varieties. (Höhentheoretische Zetafunktionen von Faserbündeln über verallgemeinerten Fahnenvarietäten.) (German) Zbl 0922.14011

Bonner Mathematische Schriften. 309. Bonn: Univ. Bonn, Mathematisch-Naturwissenschaftliche Fakultät, 43 S. (1998).
One can attach a height function \(H_L : X(K) \rightarrow R_+ \) to a metrized line bundle \(L\) on a projective variety \(X\) defined over a number field \(K\).
There is a conjecture that for a class of Fano varieties \(X\): \[ \# \{ x \in U(K) : H_L(x) \leq H\}\cong cH^a(\log H)^{b-1}, \] where \(U\) is an open subset in \(X\) and the integers \(a\) and \(b\) have a simple meaning in terms of the cone of effective divisors classes and its relation to the canonical class. The author proves the conjecture for a wide class of fibrations over the generalized flag varieties \(G/P\) with fibers of the same type. Previously, it was known for the flag varieties itself [see J. Franke, Y. I. Manin and Yu. Tschinkel, Invent. Math. 95, No. 2, 421-435 (1989; Zbl 0674.14012)].

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14M15 Grassmannians, Schubert varieties, flag manifolds
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G35 Varieties over global fields

Citations:

Zbl 0674.14012
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