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An asymptotic estimate for heights of algebraic subspaces. (English) Zbl 0773.11041

Let \(K\) be a number field and let \(S\) be a linear subspace of \(K^ n\). We define the height \(H(S)\) to be the (multiplicative, relative) Weil height of the vector of the Grassmann coordinates of \(S\), regarded as a point in projective space. Let \(M(n,d,B)\) denote the number of linear subspaces \(S\subseteq K^ n\) with \(\dim S=d\) and \(H(S)\leq B\). The main result of this paper is that, for fixed \(K\), \(n\), and \(d\), we have \(M(n,d,B)=a(n,d)B^ n+O(B^{n-b(n,d)})\) as \(B\to\infty\), where \(b(n,d)=1/([K:\mathbb{Q}]\min(d,n-d))\) and \(a(n,d)\) is given as an expression involving only \(n\), \(d\), and the regulator, class number, etc., of \(K\).
The proof proceeds by reformulating the definition of \(H(S)\) in terms of lattices over \(\mathbb{Z}\) in the adeles and then applying induction on \(n\).
Reviewer: P.Vojta (Berkeley)

MSC:

11G99 Arithmetic algebraic geometry (Diophantine geometry)
11H16 Nonconvex bodies
11J99 Diophantine approximation, transcendental number theory
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