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Asymptotic estimates for rational points of bounded height on flag varieties. (English) Zbl 0806.11030

Let \(K\) be an algebraic number field and \(G= GL_ n(K)\) the group of invertible \(n\times n\)-matrices. Each matrix in \(G\) defines a linear transformation on \(V:= K^ n\) by acting on the right to row vectors. Let \({\mathbf v}_ 1= (1,0,\dots,0), \dots,{\mathbf v}_ n\) denote the standard basis of \(V\) and let \(V_ i\) be the vector space with basis \({\mathbf v}_ 1,\dots, {\mathbf v}_ i\). For given integers \(d_ 1,\dots, d_ l\) with \(0< d_ 1<\dots <d_ l<n\), let \(P\) be the subgroup of \(G\) consisting of those linear transformations mapping the subspaces \(V_{d_ 1},\dots, V_{d_ l}\) to itself. Then \(P\setminus G\) parametrises the nested sequences of linear subspaces \(S_ 1\subset S_ 2\subset \cdots\subset S_ l\) of \(V\) with \(\dim S_ i= d_ i\) for \(i=1,\dots, l\). \(P\setminus G\) may be considered as the set of \(K\)- rational points of a projective variety \({\mathcal V}\), called a flag variety. The author defines a (non-logarithmic) height \(H\) on \({\mathcal V}(K)\) by using a suitably metrised ample line bundle (in fact the anti- canonical bundle of \({\mathcal V})\). Let \(N({\mathcal V}(K),B)\) denote the number of points of \({\mathbf x}\in {\mathcal V}(K)\) with \(H(x)\leq B\). J. Franke, Yu. I. Manin, and Yu. Tschinkel [Invent. Math. 95, 421-435 (1989; Zbl 0674.14012)] derived an asymptotic formula of the type \(N({\mathcal V}(K), B)\sim cB\log^{l-1}(B)\) as \(B\to\infty\), using results on the analytic continuation of certain \(L\)-series. The author derives a similar asymptotic formula in a much less involved way, using geometry of numbers and lattice point counting arguments. The author’s method of proof resembles that of S. H. Schanuel who derived an asymptotic formula for \(N({\mathcal V}(K), B)\) for \({\mathcal V}= \mathbb{P}^ m\) [Bull. Soc. Math. Fr. 107, 433-449 (1979; Zbl 0428.12009)].

MSC:

11G99 Arithmetic algebraic geometry (Diophantine geometry)
11H99 Geometry of numbers
14M15 Grassmannians, Schubert varieties, flag manifolds
14G05 Rational points
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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References:

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