In the literature, there are various asymptotic formulas which give, for algebraic varieties \(X\) from certain classes defined over a number field \(K\), asymptotic formulas as \(N\to\infty\) for the number of points \(x\in X(K)\) of absolute height at most \(N\). In the present paper, the author proves in certain cases analogues where the underlying field \(K\) is a function field of positive characteristic.
In the author’s set-up, \(t\) is transcendental over the finite field \(\mathbb{F}_q\), \(K\) a finite extension of genus \(g\) and degree \(\kappa\) over \(\mathbb{F}_q(t)\), and \(\overline{K}\) the algebraic closure of \(K\). It is assumed that \(\mathbb{F}_q\) is the constant field of \(K\). Define the normalized absolute height \(\overline{h}\) on \(\overline{K}^n\) or \(\mathbb{P}^{n-1}(\overline{K})\) such that if \(\mathbf{x}=(x_1,\dots , x_n)\in E^n\) for some finite extension \(E\) of \(K\) and \(M_E\) is the set of normalized discrete valuations of \(E\), then \(\overline{h}(\mathbf{x}):=-{1\over [K:\mathbb{Q}]}\sum_{v\in M_E} \min_i v(x_i)\). The absolute height of a linear subspace of \(\overline{K}^n\) is defined by taking the normalized absolute height of the exterior product of any basis of this space.
The author’s first result is an asymptotic formula as \(m\to\infty\) for the number \(\overline{N}(d,m)\) of \(\rho\in \overline{K}\) of degree \(d\) over \(K\) with height \(\overline{h}(1,\rho)=m\). This asymptotic formula gives \(\overline{N}(d,m)=c(d)q^{\kappa d(d+1)m}(1+o(1))\) as \(m\to\infty\) with a more precise error term, with \(c(d)\) a constant explicitly given in terms of \(d\) and \(K\).
Further, the author formulates a conjecture, giving an asymptotic formula as \(m\to\infty\) for the number of linear subspaces \(S\subseteq K^n\) of absolute height equal to \(m\) lying in a given Schubert cell. More precisely, let \(\mathbf{e}_1,\dots , \mathbf{e}_n\) denote the standard basis vectors of \(K^n\), and denote by \(K^d\) the subspace of \(K^n\) spanned by \(\mathbf{e}_1,\dots , \mathbf{e}_d\). Given a tuple of integers \(\mathbf{a}=(a_1,\dots , a_d)\) with \(1\leq a_1<\cdots <a_d\leq n\), denote by \(N' (\mathbf{a},m)\) the number of linear subspaces \(S\subseteq K^n\) of absolute height \(m\) such that \(\dim (S\cap K^{a_i})=i>\dim (S\cap K^{a_i-1})\) for all \(i=1,\dots ,d\). Then the author’s conjecture gives an asymptotic formula for \(N'(\mathbf{a},m)\) as \(m\to\infty\). He proves his conjecture in various cases, in particular in the case that \(K\) has genus \(0\).
The author’s proofs are based on results from his paper [J. Number Theory 128, No. 12, 2973–3004 (2008; Zbl 1216.11068)], where he considers twisted heights instead of the usual absolute height, and counts points with given twisted height.