MATHEMATICS

B. F. SKUBENKO

ON THE ASYMPTOTICS OF INTEGER MATRICES OF ORDER \(n\) AND ON THE INTEGRAL INVARIANT OF THE GROUP OF UNIMODULAR MATRICES

(Presented by Academician I. M. Vinogradov, 12 VI 1963)

Let \(N\) be a sufficiently large natural number. Let
\[ N=p_1^{k_1}p_2^{k_2}\cdots p_m^{k_m} \]
be the canonical decomposition of \(N\). Consider an integer matrix of order \(n\)
\[ X=(x_{ij}),\qquad \det X=N, \]
and associate with it the matrix
\[ \widetilde X=\left(\frac{x_{ij}}{N^{1/n}}\right)=(\widetilde x_{ij}),\qquad \det \widetilde X=1. \]

We shall denote by the symbol \(\Omega\) some Jordan-measurable domain among the unimodular matrices of order \(n\), and by the symbol \(\operatorname{mes}\Omega\) the Haar measure of this domain. Further, denote by \(D_k\) the greatest common divisor of the minors of order \(k\) of the matrix \(X\).

We shall consider the surface \(\det X=N\). On this surface we shall call a point primitive if \(D_1=1\), and completely primitive if \(D_{n-1}=1\).

Theorem 1. The number of integer points \(f(\Omega,N,n)\) on the surface \(\det X=N\) with \(\widetilde X\in\Omega\) satisfies the asymptotic formula
\[ f(\Omega,N,n)\sim \frac{\operatorname{mes}\Omega}{\zeta(2)\cdots\zeta(n)} \prod_{p_i^{k_i}} \frac{(p_i^{k_i+1}-1)(p_i^{k_i+2}-1)\cdots(p_i^{k_i+n-1}-1)} {(p_i-1)(p_i^2-1)\cdots(p_i^{n-1}-1)} \]
as \(N\to\infty\), with \(\Omega\) fixed.

This theorem is an immediate consequence of the following theorem.

Theorem 2. The number of completely primitive points \(f_{n-1}(\Omega,N,n)\) on the surface \(\det X=N\) with \(X\in\Omega\) satisfies the asymptotic formula
\[ f(\Omega,N,n)\sim \frac{\operatorname{mes}\Omega}{\zeta(2)\cdots\zeta(n)} \frac{N^n}{\varphi(N)} \sum_{d\mid N}\frac{\mu(d)}{d^n} \]
as \(N\to\infty\), with \(\Omega\) fixed.

Theorem 2 is proved with the aid of a lemma, which is also of independent interest.

Lemma. If a completely primitive matrix of order \((n-1)\)
\[ B=(b_{ij}),\qquad \det B=q, \]
is such that
\[ |b_{ij}|\le q^{\frac{1}{\,n-1\,}+\varepsilon_n},\qquad \text{o.n.d. }(q,N)\le q^{\eta_0}, \]
then the number of distinct solutions of the equation
\[ \left| \begin{array}{cc} B & \begin{pmatrix} u_1\\ \cdot\\ \cdot\\ \cdot\\ u_{n-1} \end{pmatrix}\\[6pt] (u_n\ \cdots\ u_{2(n-1)}) & t \end{array} \right| = N \]

in the variables \(u_1,\ u_2,\ldots,u_{n-1},\ u_n,\ldots,u_{2(n-1)},\ t\) for fixed \(B\), with the condition
\[ \gcd(M_{n1},M_{n2},\ldots,M_{n\,n-1},q)=1, \]
where \(M_{ij}\) is the minor of the element \(a_{ij}\) \((a_{ij}=b_{ij}\) for \(i,j<n;\ a_{ij}=u_k\) otherwise), and, for
\[ a_j q^{\frac{1}{n-1}}<u_j\leq a'_j q^{\frac{1}{n-1}},\qquad a'_j-a_j\geq q^{-\varepsilon_1},\qquad (j=1,2,\ldots,2(n-1)), \]
there is
\[ R=\varphi(q)\prod_{j=1}^{2(n-1)}(a'_j-a_j) \left(1+O\left(q^{-\left(\frac{1}{(n-1)(4n-1)}-\frac{\varepsilon_1+\eta_0}{4n-1}-\frac{4(n-1)}{4n-1}(\varepsilon_0+\varepsilon_1)-\zeta\right)}\right)\right), \]
where \(\zeta\) is an arbitrarily small positive quantity; the constant in the remainder depends only on \(\zeta\).

In the course of proving this lemma, an interesting arithmetical observation emerges.

If \(\gcd(q,N)=1\), then \(\gcd(q,M_{nj})\) and \(\gcd(q,M_{in})\) \((i=1,2,\ldots,n-1)\) are invariants of the solutions of the indeterminate equation
\[ \left| \begin{array}{ccccc} b_{11} & b_{12} & \ldots & b_{1\,n-1} & u_1\\ b_{21} & b_{22} & \ldots & b_{2\,n-1} & u_2\\ \cdot & \cdot & \cdot & \ldots & \cdot\\ \cdot & \cdot & \cdot & \ldots & \cdot\\ \cdot & \cdot & \cdot & \ldots & \cdot\\ b_{n-1\,1} & b_{n-1\,2} & \ldots & b_{n-1\,n-1} & u_{n-1}\\ u_n & u_{n+1} & \ldots & u_{2(n-1)} & t \end{array} \right|=N \]
in the variables \(u_1,u_2,\ldots,u_{n-1},u_n,\ldots,u_{2(n-1)},t\).

Our lemma is proved with the aid of the well-known lemma of I. M. Vinogradov on Fourier series\({}^{(1)}\) and estimates of A. Weil\({}^{(2)}\).

It follows from Theorem 1 that the volume \(V_n\) of the quotient space of the group of unimodular matrices by the subgroup of integral unimodular matrices of order \(n\) is
\[ V_n=\zeta(2)\zeta(3)\cdots\zeta(n). \]

For the proof it suffices to note that: 1) in the space of the unimodular group one can always choose a fundamental domain so that it is quadrable in the sense of Jordan; 2) the expression
\[ \prod_{p_i^{k_i}} \frac{(p_i^{k_i+1}-1)(p_i^{k_i+2}-1)\cdots(p_i^{k_i+n-1}-1)} {(p_i-1)(p_i^2-1)\cdots(p_i^{n-1}-1)} \]
is nothing other than the number of left (right) non-associated integral matrices of order \(n\) with determinant \(N\).

Received
3 VI 1963

References

\({}^{1}\) I. M. Vinogradov, Selected Works, 1952.
\({}^{2}\) A. Weil, Proc. Nat. Akad. USA, 34, 204 (1948).