L. A. GUTNIK

ON THE ARITHMETIC OF MATRICES

(Presented by Academician I. M. Vinogradov on 8 IV 1958)

Let the group of all integral matrices of order (n) with determinant congruent modulo to 1 be denoted by (U_n). A diagonal matrix (\delta) of order (n) is called an (ET)-matrix if it is diagonal,

[
\delta_{i;i}>0,\quad i=1,\ldots,n;\qquad
\delta_{i;i}\mid \delta_{i+1;i+1},\quad i=1,\ldots,n-1.
\tag{1}
]

The determinant of a matrix (\alpha) is denoted by (D(\alpha)); the matrix transposed to (\alpha) is denoted by (\alpha'). For denoting zero matrices the letter (\omega) is reserved, and for denoting identity matrices—the letter (\varepsilon). Let (\alpha) be a matrix of order (n); let (\varepsilon) and (\omega) be, respectively, the identity and zero matrices of order (n). The symbol (\alpha^A) denotes the matrix
[
\begin{pmatrix}
\omega & \alpha\
-\alpha' & \omega
\end{pmatrix},
]
and the symbol (\alpha^B) denotes the matrix
[
\begin{pmatrix}
\varepsilon & \omega\
\omega & \alpha
\end{pmatrix}.
]
Let (\alpha) be an arbitrary integral nonsingular matrix of order (n). Using the theorem on elementary divisors, it is easy to show that in the set (U_n\alpha U_n) there lies one and only one (ET)-matrix; this (ET)-matrix, invariantly associated with the matrix (\alpha), is called the (E)-form of the matrix (\alpha) and is denoted by (\alpha^E). Let (\pi) be an integral nonsingular skew-symmetric matrix of order (2n). From the well-known theorem of Jacobi it follows that in the set (U_{2n}\pi U_{2n}) there lies one and only one matrix of the form (\delta^A), where (\delta) is an (ET)-matrix; this matrix of the form (\delta^A), invariantly associated with the matrix (\pi), is called the (ES)-form of the matrix (\pi) and is denoted by (\pi^{ES}).

Let (\alpha) be an integral nonsingular matrix of order (n). An integral nonsingular matrix (\beta) is called an element of a left (right) divisor of the matrix (\alpha), if the matrix (\beta^{-1}\alpha) (the matrix (\alpha\beta^{-1})) is integral. If (\beta) is an element of a left (right) divisor of the matrix (\alpha), then (see (3)) (\beta^E\mid \alpha^E). If (\beta) is an element of a left (right) divisor of the matrix (\alpha), (\nu\in\Omega_n), then the matrix (\beta\nu) (the matrix (\nu\beta)) is also an element of a left (right) divisor of the matrix (\alpha). Therefore the set of matrices associated on the right (on the left) with respect to the group (U_n) with some element of a left (right) divisor of the matrix (\alpha) consists entirely of elements of a left (right) divisor of the matrix (\alpha); this set we shall call a left (right) divisor of the matrix (\alpha). It is easily proved that the set of left (right) divisors of the matrix (\alpha) is finite and contains no more than (D^n(\alpha)) elements. The number of left (right) divisors of the matrix (\alpha) will be denoted by (t_l(\alpha)) (by (t_r(\alpha))). It is obvious that the matrices belonging to a certain left (right) divisor of the matrix (\alpha) have one and the same (E)-form. The number of left (right) divisors of the matrix (\alpha) having one and the same (E)-form (\gamma) will be denoted by (t_l(\alpha,\gamma)) (by (t_r(\alpha,\gamma))). From (1) it follows that

[
t_l(\alpha)=\sum_{\gamma\mid\alpha^E} t_l(\alpha,\gamma),\qquad
t_r(\alpha)=\sum_{\gamma\mid\alpha^E} t_r(\alpha,\gamma).
\tag{2}
]

Using the operation of transposition, it is easy to show that

[
t_l(\alpha,\gamma)=t_r(\alpha',\gamma).
\tag{3}
]

Further, it is not difficult to show that

[
t_l(\alpha,\gamma)=t_l(\alpha^E,\gamma),\qquad
t_r(\alpha,\gamma)=t_r(\alpha^E,\gamma).
\tag{4}
]

Comparing (3) and (4), we obtain

[
t_l(\alpha,\gamma)=t_r(\alpha,\gamma),\qquad
t_l(\alpha)=t_r(\alpha).
\tag{5}
]

In view of (5), put (t(\alpha,\gamma)=t_l(\alpha,\gamma)=t_r(\alpha,\gamma)), (t(\alpha)=t_l(\alpha)=t_r(\alpha)). Then (2) gives

[
t(\alpha)=\sum_{\gamma\mid\alpha^E} t(\alpha,\gamma).
\tag{6}
]

It is natural to call the function (t(\alpha)) the number of divisors of the matrix (\alpha): in the case when (\alpha) is a matrix of the first order, (t(\alpha)) is equal to the number of divisors of the modulus of the number (\alpha). In view of (4), in studying the functions (t(\alpha)) and (t(\alpha,\gamma)) one may restrict oneself to matrices (\alpha) that are (ET)-matrices.

Let (a) be a nonzero integer; (p) a prime number; by (a^{[p]}) denote the highest integral power of (p) dividing (a). Let (\delta) be an (ET)-matrix of order (n); (p) a prime number; by (\delta^{[p]}) denote the (ET)-matrix having diagonal elements (\delta^{[p]}{1;1},\ldots,\delta^{[p]}}). Then every (ET)-matrix (\delta) is represented uniquely in the form (\delta=\prod_{p\mid D(\delta)}\delta^{[p]}). Let (\gamma) and (\delta) be (ET)-matrices of order (n); (p) a prime number. Introduce the following symbols: (e^{[p]}(\gamma)) is the number of elements of the sequence (\gamma^{[p]{1;1},\ldots,\gamma^{[p]}}) equal to 1; (\mathrm{Б}^{[p]i(\delta,\gamma)) is the number of elements of the sequence (\delta^{[p]}},\ldots,\delta^{[p]{n;n}) not smaller than (\gamma^{[p]}}); (\mathrm{Ц}^{[p]i(\delta,\gamma)) is the number of elements of the sequence (\delta^{[p]}},\ldots,\delta^{[p]{n;n}) greater than (\gamma^{[p]}}); (\mathrm{Э}^{[p]i(\delta,\gamma)) is the number of elements of the sequence (\delta^{[p]}).}), (i\leq j\leq n), equal to (\gamma^{[p]}_{i;i

Theorem 1.

[
t(\delta,\gamma)=
]

[
=\prod_{p\mid D(\delta)}\prod_{i=1}^{n}
\left(\gamma^{[p]}{i;i}\right)^{-\mathrm{Б}^{[p]}_i(\delta,\gamma)+\mathrm{E}^{[p]}_i(\gamma,\gamma)-\mathrm{Ц}^{[p]}_i(\gamma,\gamma)}
\left(\delta^{[p]}}\right)^{\mathrm{Ц}^{[p]}_i(\gamma,\delta)
\frac{1-p^{\mathrm{Б}^{[p]}_i(\delta,\gamma)-i}}
{1-p^{\mathrm{Б}^{[p]}_i(\gamma,\gamma)-i}}.
\tag{7}
]

Let (\pi,\rho) be two integral, nonsingular, skew-symmetric matrices of order (2n). Denote by (A_0(\pi,\rho)) the set of all integral matrices (\chi) satisfying the relation (\chi'\pi\chi=\rho). Clearly, the set (A_0(\pi,\pi)) is a group. If (\mu\in A_0(\pi,\pi)), (\nu\in A_0(\rho,\rho)), then (\mu A_0(\pi,\pi)\nu=A_0(\rho,\rho)). The quotient space of the set (A_0(\pi,\rho)) with respect to the group of automorphisms (A_0(\pi,\rho)\to \mu A_0(\pi,\rho)), (\mu\in A_0(\pi,\pi)), will be denoted by (L_0(\pi,\rho)). The quotient space of the set (A_0(\pi,\rho)) with respect to the group of automorphisms (A_0(\pi,\rho)\to A_0(\pi,\rho)\nu), (\nu\in A_0(\rho,\rho)), will be denoted by (R_0(\pi,\rho)). It is easily proved that the sets (L_0(\pi,\rho)) and (R_0(\pi,\rho)) are finite and that the number of elements of each of them does not exceed (\left(D(\rho\pi^{-1})\right)^{2n^2}). Denote the number of elements of the set (L_0(\pi,\rho)) by (a_l(\pi,\rho)), and the number of elements of the set (R_0(\pi,\rho)) by (a_r(\pi,\rho)). Then it is not difficult to show that (a_l(\pi,\rho)=a_l(\pi^{ES},\rho^{ES})), (a_r(\pi,\rho)=a_r(\pi^{ES},\rho^{ES})). Therefore, in studying the numbers (a_l(\pi,\rho)) and (a_r(\pi,\rho)) one may restrict oneself to matrices (\pi) and (\rho) having the form (\pi=\gamma^A), (\rho=\delta^A), where (\gamma) and (\delta) are (ET)-matrices.

Theorem 2. Let (\varepsilon) be the identity matrix of order (n), and let (\delta) be an (ET)-matrix of order (n). Then

[
a_l(\varepsilon^A,\delta^A)=
\sum_{\gamma\mid\delta} t(\delta,\gamma)\prod_{i=1}^{n}\gamma_{i;i}^{\,n-i+1},
]

where the sum is extended over all (ET)-matrices (\gamma) for which the matrix (\gamma^{-1}\delta) is integral.

Let (A) be a group, and (B) a subgroup of finite index of the group (A). By ([A:B]) we denote the index of the subgroup (B) in the group (A). Put
(A(\gamma,\delta)=\gamma^B A_0(\gamma^A,\delta^A)(\delta^B)^{-1}), (M(\gamma)=A(\gamma,\gamma)). It is easy to see that (M(\gamma)) is a subgroup of the symplectic group. As was shown by Kézher ((^4)),

[
\frac{a_r(\gamma^A,\delta^A)}{a_l(\gamma^A,\delta^A)}
=
\frac{[M(\gamma):K(\gamma,\delta)]}{[M(\delta):K(\gamma,\delta)]},
]

where (K(\gamma,\delta)=M(\gamma)\cap M(\delta)).

Theorem 3.

[
\frac{[M(\gamma):K(\gamma,\delta)]}{[M(\delta):K(\gamma,\delta)]}
=
\prod_{p\mid D(\gamma\delta)}
\frac{
\prod_{i=1}^{n}(\gamma_{i;i}^{[p]})^{2n-4i+2}
\left(1-\frac{1}{p^{\mathfrak e_i^{[p]}(\gamma,\gamma)}}\right)
}{
\prod_{i=1}^{n}(\delta_{i;i}^{[p]})^{2n-4i+2}
\left(1-\frac{1}{p^{\mathfrak e_i^{[p]}(\delta,\delta)}}\right)
}.
\tag{9}
]

Let (\pi) and (\rho) be two integral nonsingular skew-symmetric matrices of order (2n); (q_1) and (q_2) two natural numbers. We divide the set of matrices satisfying the congruence (\chi'\pi\chi\equiv \rho \pmod {q_1}) into classes, assigning to one class matrices congruent to one another modulo (q_2). We denote the number of classes obtained by (N_{q_1,q_2}(\pi,\rho)).

Theorem 4. Let (\delta) be an (ET)-matrix of order (n); (p) a prime number; (b) a natural number satisfying the requirement

[
p^b>\delta_{n;n}^{[p]}.
\tag{10}
]

Then

[
N_{p^b,p^b}(\delta^A,\delta^A)
=
p^{bn(2n+1)}
\prod_{i=1}^{n}
(\delta_{i;i}^{[p]})^{4n-4i+1}
\left(1-\frac{1}{p^{\mathfrak e_i^{[p]}(\delta,\delta)}}\right).
\tag{11}
]

For (\delta=\varepsilon), from formula (10) there follows the well-known fact

[
N_{p^b,p^b}(\varepsilon^A,\varepsilon^A)
=
p^{bn(2n+1)}
\prod_{i=1}^{n}\left(1-\frac{1}{p^{2i}}\right).
\tag{12}
]

Theorem 5. Let (\delta) be an (ET)-matrix of order (n); (p) a prime number; (b,c) natural numbers subject to the requirement (p^b(\delta_{n;n}^{[p]})^{-1}>p^c>\delta_{n;n}^{[p]}). Then

[
N_{p^b,p^c}(\delta^A,\delta^A)
=
p^{cn(2n+1)}
\prod_{i=1}^{n}
(\delta_{i;i}^{[p]})^{2n-4i+2}
\left(1-\frac{1}{p^{\mathfrak e_i^{[p]}(\delta;\delta)}}\right).
\tag{13}
]

From formula (11) it follows that, when inequality (10) is satisfied, the quantity
(N_{p^b,p^b}(\delta^A,\delta^A)/p^{bn(2n+1)}) does not depend on (b).

Using an idea of Siegel, one can prove the following general fact.

Theorem 6. Let (\pi,\rho) be two integral nonsingular matrices of order (2n); (p) a prime number; (p^{b_0}) the highest integral power entering into the determinant of the matrix (\rho); (b) an integer satisfying the requirement (b>2b_0). Then the number (N_{p^b,p^b}(\pi,\rho)p^{-bn(2n+1)}) does not depend on the number (b).

We denote this number independent of (b) by (c_p(\pi,\rho)). Let (\rho) be a nonsingular skew-symmetric matrix of order (2n). By (\Psi(\rho)) we denote some neighborhood of the matrix (\rho), consisting of skew-symmetric matrices of order (2n). If (\psi\in\Psi(\rho)), then the elements (\psi_{k,l}), (1\le k<l\le 2n), are regarded as independent rectangular coordinates in an (n(2n-1))-dimensional Euclidean space. Let (\pi) also be some nonsingular skew-symmetric matrix of order (2n). By (\Omega(\pi,\Psi)) we denote the set of all real matrices satisfying the requirement (\xi'\pi\xi\in\Psi).

If (\xi\in\Omega(\pi,\Psi)), the elements (\xi_{i,j}), (i,j=1,\ldots,2n), will be regarded as independent rectangular coordinates in (4n^2)-dimensional Euclidean space. It is easy to see that if (\mu\in A_0(\pi,\pi)), then
[
\mu\Omega(\pi,\Psi)=\Omega(\pi,\Psi).
]
It can be proved that if the diameter of the domain (\Psi(\rho)) is sufficiently small, then there exists a fundamental domain (\Omega_0(\pi,\Psi)) of the space (\Omega(\pi,\Psi)) with respect to its automorphism group (\Omega(\pi,\Psi)\to \mu\Omega(\pi,\Psi)), (\mu\in A_0(\pi,\pi)), having finite volume. Let (M) be some set having finite volume; then the volume of the set (M) is denoted by (v(M)). It can be shown that if the neighborhood (\Psi(\rho)) is contracted to the point (\rho), then the ratio
[
v(\Omega_0(\pi,\Psi))/v(\Psi(\rho))
]
tends to a certain finite nonzero limit, which we denote by (c_0(\pi,\rho)). Since, if (p) does not divide (D(\pi\rho)), then
[
c_p(\pi,\rho)=\prod_{i=1}^{n}\left(1-\frac{1}{p^{2i}}\right),
]
it is easy to see that the product
[
\prod_{(p,D(\pi\rho))=1} c_p(\pi,\rho),
]
extended over all prime numbers not entering into the determinant (D(\pi,\rho)), converges. We denote by
[
\prod_p c_p(\pi,\rho)
]
the quantity
[
\prod_{(p,D(\pi\rho))=1} c_p(\pi,\rho)\prod_{p\mid D(\pi\rho)} c_p(\pi,\rho).
]
In Theorem 7 there is proved a statement asserted by I. I. Pjateckii-Shapiro.

Theorem 7.
[
a_1(\pi,\rho)=c_0(\pi,\rho)\prod_p c_p(\pi,\rho).
\tag{14}
]

The proof of Theorem 7 is easily reduced to the case when (\pi=\gamma^A), (\rho=\delta^A). Siegel ({}^{(2)}) showed that
[
1=c_0(\varepsilon^A,\varepsilon^A)\prod_p c_p(\pi,\rho).
\tag{15}
]
Using (9), we find an expression for (c_0(\delta^A,\delta^A)/c_0(\varepsilon^A,\varepsilon^A)); comparing it with (11), (12), and (15), we obtain the proof of Theorem 7 for the case when (\pi=\rho); from the result thus obtained, applying Siegel’s method, developed by him in ({}^{(1)}), we derive relation (14). Relation (14) may be regarded as an analogue of Siegel’s theorem on quadratic forms ({}^{(1)}). From Theorems 7, 3, and 6 follow Theorems 8 and 9.

Theorem 8. Let (\gamma,\delta) be matrices of order (n). Then
[
a_1(\gamma^A,\delta^A)=\prod_{p\mid D(\gamma\delta)} a_1\bigl((\gamma^{[p]})^A,(\delta^{[p]})^A,(\rho^{[p]})^A\bigr),
]
[
a_r(\gamma^A,\delta^A)=\prod a_r\bigl((\gamma^{[p]})^A,(\delta^{[p]})^A\bigr).
]

Theorem 9. Let (\pi,\rho) be two nonsingular skew-symmetric matrices of order (2n); let (q) be a natural number; (D^3(\pi\rho)\backslash q). Then an integral matrix (\chi) satisfying the relation
[
\chi'\pi\chi=\rho
]
exists if and only if the congruence
[
\chi'\pi\chi\equiv\rho\pmod q
]
is solvable.

Theorem 9 may be regarded as an analogue of the well-known Hasse theorem on quadratic forms.

In conclusion I express my sincere gratitude to Prof. A. A. Buchstab for his attention to the present work.

Received
3 IV 1958

References

({}^{1}) C. L. Siegel, Ann. Math., 36, 527 (1935).
({}^{2}) C. L. Siegel, Am. J. Math., 65, 1, (1943).
({}^{3}) M. Koecher, Math. Nachr., 13, 367 (1955).
({}^{4}) M. Koecher, Math. Ann., 130, H. 5, 351, (0a) (1956).