Georgian Mathematical Journal Volume 9 (2002), Number 1, 167–178

ON THE DIMENSION OF SOME SPACES OF GENERALIZED TERNARY THETA-SERIES

K. SHAVGULIDZE

Abstract. The upper bound of dimension of vector spaces of generalized theta-series corresponding to some ternary quadratic forms is established. In a number of cases, the dimension of vector spaces of generalized theta-series is established and bases of these spaces are constructed.

2000 Mathematics Subject Classification: 11E20, 11F27, 11F30.

Key words and phrases: Positive reduced ternary quadratic form, spherical polynomial with respect positive quadratic form, vector space of generalized theta-series, integral automorph of a quadratic form.

1. Introduction

Let

Q(X) = Q(x_{1}, …, x_{r}) = ∑_{1≤i≤j≤r} b_{ij}x_{i}x_{j}

be an integral positive definite quadratic form in an even number r of variables. That is, b_{ij} ∈ ℤ and Q(X) > 0 if X ≠ 0. To Q(X) we associate the even integral symmetric r × r matrix A defined by a_{ii} = 2b_{ii} and a_{ij} = a_{ji} = b_{ij}, where i < j. If X = [x_{1}, …, x_{r}]′ denotes a column vector where ′ denotes the transposition, then we have Q(X) = \frac{1}{2}X′AX. Let A_{ij} denote the algebraic adjunct of the element a_{ij} in D = det A and a_{ij}^{∗} the corresponding element of A^{−1}.

Below we shall use the notions, notation and some results from [2].

A homogeneous polynomial P (X) = P (x_{1}, …, x_{r}) of degree ν with complex coefficients, satisfying the condition

∑_{1≤i,j≤r} a_{ij}^{∗}(\frac{∂^{2}P}{∂x_{i}∂x_{j}} ) = 0  (1)

is called a spherical polynomial of order ν with recpect to Q(X) (see [1]), and

ϑ(τ, P, Q) = ∑_{n∈ℤ^{∖}} P (n)z^{Q(n)}, z = e^{2πiτ}, τ ∈ ℂ, Im τ > ⊬,  (2)

is the corresponding generalized r-fold theta-series.

Let R(ν, Q) denote the vector space over ℂ of spherical polynomials P (X) of even order ν with respect to Q(X). Hecke [2] calculated the dimension of the space R(ν, Q) and showed that

dim R(ν, Q) = \frac{(ν + ∇ − ∋)!}{(∇ − ∈)!ν!} (∇ + ∈ν − ∈).  (3)

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Let T (ν, Q) denote the vector space over ℂ of generalized multiple thetaseries, i.e.,

T (ν, Q) = {ϑ(τ, P, Q) : P ∈ R(ν, Q)}.

Gooding [1] calculated the dimension of the vector space T (ν, Q) for reduced binary quadratic forms Q.

In this paper the upper bound is established for dimension of the vector space T (ν, Q) for some positive reduced ternary quadratic forms. In a number of cases the dimension of vector spaces of theta-series is also calculated and the bases of this spaces are constructed.

In the sequel we use the following definition and results:

An integral r × r matrix U is called an integral automorph of the quadratic form Q(X) in r variables if the condition

U′AU = A  (4)

is satisfied.

Lemma 1 ([1], p. 37). Let Q(X) = Q(x_{1}, …, x_{r}) be a positive definite quadratic form in r variables and P (X) = P (x_{1}, …, x_{r}) ∈ R(ν, Q). Let G be the set of all integral automorphs of Q. Suppose

∑_{i=1}^{t} P (U_{i}X) = 0 for some U_{1}, …, U_{t} ⊆ G,

then ϑ(τ, P, Q) = 0.

Lemma 2 ([2], p. 853). Among homogeneous quadratic polynomials in r variables

φ_{ij} = x_{i}x_{j} − \frac{A_{ij}}{rD} 2Q(X) (i, j = 1, …, r),  (5)

exactly \frac{r(r+1)}{2} − 1 ones are linearly independent and form the basis of the space of spherical polynomials of second order with respect to Q(X).

Lemma 3 ([3], p. 533). Among homogeneous polynomials of fourth degree in r variables

φ_{ijkl} = x_{i}x_{j}x_{k}x_{l} − \frac{1}{(r + 4)D} (A_{ij}x_{k}x_{l} + A_{ik}x_{j}x_{l} + A_{il}x_{j}x_{k} + A_{jk}x_{i}x_{l}

+ A_{jl}x_{i}x_{k} + A_{kl}x_{i}x_{j}) · 2Q + \frac{1}{(r + 2)(r + 4)D^{2}} · (A_{ij}A_{kl} + A_{ik}A_{jl}

+ A_{il}A_{jk}) · (2Q)^{2} (i, j, k, l = 1, …, r),  (6)

exactly \frac{1}{24} r(r^{2} − 1)(r + 6) ones are linearly independent and form the basis of the space of spherical polynomials of fourth order with respect to Q(X).

Lemma 4 ([4], p. 28–30). For a matrix of integers

U = ∥u_{ij}∥_{r×r}

to be an integral automorph of the quadratic form

Q(X) = ∑_{1≤i≤j≤r} b_{ij}x_{i}x_{j}

it is sufficient

a) to find a representation of integers b_{11}, b_{22}, …, b_{rr} by the quadratic form Q(X), assuming that

Q(u_{1i}, u_{2i}, …, u_{ri}) = b_{ii} (1 ≤ i ≤ r),  (7)

and b) to verify that the \frac{r^{2}−r}{2} conditions

∑_{t=1}^{r} ∑_{k=1}^{r} a_{tk}u_{ti}u_{kj} = b_{ij} (i < j; i, j = 1, …, r)  (8)

are satisfied.

2. The Basis of the Space R(ν, Q)

Let

P (x) = P (x_{1}, x_{2}, x_{3}) = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k}  (9)

be a spherical function of order ν with respect to the positive ternary quadratic form Q(x_{1}, x_{2}, x_{3}). Hence, according to definition (1), the condition

\frac{A_{11}}{|A|} \frac{∂^{2}P}{∂x_{1}^{2}} + 2 \frac{A_{12}}{|A|} \frac{∂^{2}P}{∂x_{1}∂x_{2}} + 2 \frac{A_{13}}{|A|} \frac{∂^{2}P}{∂x_{1}∂x_{3}} + \frac{A_{22}}{|A|} \frac{∂^{2}P}{∂x_{2}^{2}}

+2 \frac{A_{23}}{|A|} \frac{∂^{2}P}{∂x_{2}∂x_{3}} + \frac{A_{33}}{|A|} \frac{∂^{2}P}{∂x_{3}^{2}} = 0  (10)

is satisfied. Since

\frac{∂^{2}P}{∂x_{1}^{2}} = ∑_{k=0}^{ν} ∑_{i=0}^{k} i(i − 1)a_{ki}x_{1}^{i−2}x_{2}^{k−i}x_{3}^{ν−k}

= ∑_{k=1}^{ν−1} ∑_{i=0}^{k−1}(i + 1)(i + 2)a_{k+1,i+2}x_{1}^{i}x_{2}^{k−i−1}x_{3}^{ν−k−1}

and one can easily obtain similar formulas for other second partial derivatives, condition (10) takes the form

\frac{1}{|A|} ∑_{k=1}^{ν−1} ∑_{i=0}^{k−1} (A_{11}(i + 1)(i + 2)a_{k+1,i+2} + 2A_{12}(k − i)(i + 1)a_{k+1,i+1}

+2A_{13}(ν − k)(i + 1)a_{k,i+1} + A_{22}(k − i)(k − i + 1)a_{k+1,i}

+2A_{23}(ν − k)(k − i)a_{ki} + A_{33}(ν − k)(ν − k + 1)a_{k−1,i})x_{1}^{i}x_{2}^{k−i−1}x_{3}^{ν−k−1} = 0.

Thus, for 0 ≤ i < k ≤ ν − 1, we obtain

A_{11}(i + 1)(i + 2)a_{k+1,i+2} + 2A_{12}(k − i)(i + 1)a_{k+1,i+1}
+2A_{13}(ν − k)(i + 1)a_{k,i+1} + A_{22}(k − i)(k − i + 1)a_{k+1,i}
+2A_{23}(ν − k)(k − i)a_{ki} + A_{33}(ν − k)(ν − k + 1)a_{k−1,i} = 0.  (11)

Let

L = [a_{00}, a_{10}, a_{11}, a_{20}, a_{21}, a_{22}, …, a_{νν}]′

be the column vector, where a_{ki} (0 ≤ i ≤ kν) are the coefficients of polynomial (9).

Conditions (11) in matrix notation have the form

S · L = 0,  (12)

where the matrix S (the elements of this matrix are defined from conditions (11)) has the form

∥ \begin{matrix}A_{33}(ν−1)ν & 2A_{23}(ν−1) & 2A_{13}(ν−1) & 2A_{22} & ... & 0 & 0 \\ 0 & A_{33}(ν−2)(ν−1) & 0 & 4A_{23}(ν−2) & ... & 0 & 0 \\ 0 & 0 & A_{33}(ν−2)(ν−1) & 0 & ... & 0 & 0 \\ ... & ... & ... & ... & ... & ... & ... \\ 0 & 0 & 0 & 0 & ... & 2A_{12}(ν−1) & A_{11}(ν−1)ν \end{matrix} ∥ .

The number of rows of the matrix S is equal to the number of conditions (11), i.e., to the number of pairs (i, k) with 0 ≤ i < k ≤ ν − 1. Hence the number of rows is equal to \binom{ν}{2} = ν(ν − 1)/2. As for the number of columns of the matrix S it is equal to the number of coefficients a_{ki} of polynomial (9), i.e., to the number of pairs (i, k) with 0 ≤ i ≤ k ≤ ν. Hence it is equal to \binom{ν+2}{2}.

We partition the matrix S into two matrices S_{1} and S_{2}. S_{1} is the left square nondegenerate \binom{ν}{2}×\binom{ν}{2} matrix, it consists of the first \binom{ν}{2} columns of the matrix S; S_{2} is the right \binom{ν}{2} × (2ν + 1) matrix, it consists of the last 2ν + 1 columns of the matrix S.

Similarly, we partition the matrix L into two matrices L_{1} and L_{2}. L_{1} is the \binom{ν}{2} × 1 matrix, it consists of the upper \binom{ν}{2} elements of the matrix L; L_{2} is the (2ν + 1) × 1 matrix, it consists of the lower 2ν + 1 elements of the matrix L.

According to the new notation, the matrix equality (12) has the form:

S_{1}L_{1} + S_{2}L_{2} = 0,

i.e.,

L_{1} = −S_{1}^{−1}S_{2}L_{2}.  (13)

It follows from this equality that the matrix L_{1} is expressed through the matrix L_{2}, i.e., the first \binom{ν}{2} elements of the matrix L are expressed through its other 2ν + 1 elements. Since the matrix L consists of the coefficients of the spherical polynomial P (X), its first \binom{ν}{2} coefficients can be expressed through the last 2ν + 1 coefficients.

Let Q(X) = Q(x_{1}, x_{2}, x_{3}) be a ternary quadratic form. According to the formula (3), dim R(ν, Q) = ∈ν + ∞.

Let us show that the polynomials^{1}

\begin{matrix}P_{1}(a_{00}^{(1)}, a_{10}^{(1)}, …, a_{ν−2,ν−2}^{(1)}, 1, 0, 0, …, 0), \\ P_{2}(a_{00}^{(2)}, a_{10}^{(2)}, …, a_{ν−2,ν−2}^{(2)}, 0, 1, 0, 0, …, 0), \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ P_{2ν+1}(a_{00}^{(2ν+1)}, a_{10}^{(2ν+1)}, …, a_{ν−2,ν−2}^{(2ν+1)}, 0, 0, 0, …, 1), \end{matrix}  (14)

where in view of (13) the first \binom{ν}{2} coefficients from a_{00} to a_{ν−2,ν−2} are calculated through other 2ν + 1 coefficients and form the basis of the space R(ν, Q).

Indeed, these polynomials satisfy condition (13), i.e., condition (1), hence polynomials (14) are spherical functions with respect to Q(X). They are linearly independent and altogether are 2ν + 1, as it is stated above that the dimension of the space R(ν, Q) is equal to 2ν + 1. Thus polynomials (14) form the basis of the space R(ν, Q).

3. The Upper Bound of Dimension of T (ν, Q) for Some Ternary Forms

Consider the quadratic form

Q_{1} = b_{11}x_{1}^{2} + b_{22}x_{2}^{2} + b_{33}x_{3}^{2},

where 0 < b_{11} < b_{22} < b_{33}.

Construct the integral automorphs U of the quadratic form Q_{1}. According to the definition (4) and Lemma 4,

b_{11} = Q(±1, 0, 0),
b_{22} = Q(0, ±1, 0),
b_{33} = Q(0, 0, ±1).

Hence it is easy to verify that the integral automorphs of the quadratic form Q_{1} are only

U = ∥\begin{matrix}e_{1} & 0 & 0 \\ 0 & e_{2} & 0 \\ 0 & 0 & e_{3} \end{matrix}∥ (e_{i} = ±1, i = 1, 2, 3).  (15)

From these 8 automorphs U we use only

U_{1} = ∥\begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}∥ and U_{2} = ∥\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix}∥ .  (16)

Consider all possible polynomials P_{h}(U_{j}X) (h = 1, 2, …, 2ν + 1), where P_{h} ∈ R(ν, Q_{∞}) are spherical polynomials of order ν with recpect to Q_{1}(X), and U_{j} ∈ G are integral automorphs of the quadratic form Q_{1}(X).

1In the brackets of (14) the coefficients of polynomial P_{i} of form (9) are shown.

From (9) and (16) it follows that

P_{h}(U_{1}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}(−x_{1})^{i}x_{2}^{k−i}x_{3}^{ν−k} = ∑_{k=0}^{ν} ∑_{i=0}^{k}(−1)^{i}a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k},

P_{h}(U_{2}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}(−x_{3})^{ν−k} = ∑_{k=0}^{ν} ∑_{i=0}^{k}(−1)^{k}a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k}.

Further, we have

P_{h}(X) + P_{h}(U_{1}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k}(1 + (−1)^{i})a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k},

P_{h}(X) + P_{h}(U_{2}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k}(1 + (−1)^{k})a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k}.

We shall find out for which P_{h}, of two equalities

P_{h}(X) + P_{h}(U_{1}X) = 0  (17)

or

P_{h}(X) + P_{h}(U_{2}X) = 0,  (18)

at least one takes place. Equality (17) takes place if and only if the coefficients

(1 + (−1)^{i})a_{ki}^{(h)} = 0  (19)

for all k, i. Keeping in mind the construction of a basis of the space of spherical functions, it is sufficient to show that (19) is true for the last 2ν + 1 coefficients from a_{ν−1,0}^{(h)} to a_{νν}^{(h)}, i.e., when k = ν − 1, ν; i = 0, 1, …, k. All these last coefficients a_{ki}^{(h)} are zero, except one which is equal to one. Suppose a_{ki}^{(h)} = 1 for a certain pair k, i with ν − 1 ≤ k ≤ ν and 0 ≤ i ≤ k; if i is odd, then (1 + (−1)^{i})a_{ki}^{(h)} = 0. Hence P_{h} satisfies equality (17) if among the last 2ν + 1 coefficients, the index i – of the coefficient equal to one is odd. Similarly, it follows that P_{h} satisfies equality (18) if among the last 2ν + 1 coefficients, the index k – of the coefficient equal to one is odd.

Calculate how many polynomials P_{h} satisfy at least one of two equalities (17) or (18), i.e., calculate how many coefficients a_{ki} with ν − 1 ≤ k ≤ ν and 0 ≤ i ≤ k have among the indices k or i, at least one odd. We have the following cases:

a) If 2 ∤ k, then k = ν − 1. The total number of coefficients with k = ν − 1 is equal to ν.

b) If 2 | k, but 2 ∤ i, then k = ν, 0 ≤ i ≤ ν, 2 ∤ i. The total number of

coefficients with such indices is equal to ν/2. Thus, altogether we have ν + ν/2 polynomials P_{h} which satisfy at least one

of equalities (17) or (18), but for such polynomials, according to Lemma 1,

ϑ(τ, P_{h}, Q_{1}) = 0.  (20)

We have thus shown that among 2ν +1 theta-series, corresponding to linearly independent spherical polynomials, ν + ν/2 theta-series are zero. Hence the maximal number of linearly independent theta-series, i.e.,

dim T (ν, Q_{1}) ≤ 2ν + 1 − (ν + \frac{ν}{2}) = \frac{ν}{2} + 1.  (21)

Now consider the quadratic form

Q_{2} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2},

where 0 < b_{11} < b_{33}.

We construct the integral automorphs U of the quadratic form Q_{2}. Since

b_{11} = Q_{2}(±1, 0, 0) = Q_{2}(0, ±1, 0), b_{33} = Q_{2}(0, 0, ±1),

it is easy to verify that the integral automorphs of the quadratic form Q_{2} are automorphs (15) and, moreover, the automorphs

∥\begin{matrix}0 & e_{1} & 0 \\ e_{2} & 0 & 0 \\ 0 & 0 & e_{3} \end{matrix}∥ (e_{i} = ±1, i = 1, 2, 3).  (22)

From automorphs (22) we use only

U_{3} = ∥\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}∥ .  (23)

The automorphs of the quadratic form Q_{1} are also automorphs of the quadratic form Q_{2}, therefore

dim T (ν, Q_{2}) ≤ \frac{ν}{2} + 1.

We improve this estimation. From (9) and (23) it follows that

P_{h}(U_{3}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}x_{2}^{i}x_{1}^{k−i}x_{3}^{ν−k} = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}x_{1}^{k−i}x_{2}^{k−(k−i)}x_{3}^{ν−k}.

From here it follows that if all last 2ν + 1 coefficients of the basis polynomial P_{h}(X) are equal to zero, except one a_{ki}^{(h)} = 1, then all last 2ν + 1 coefficients of polynomial P_{h}(U_{3}X) are equal to zero, except one a_{k,k−i} = 1. Hence P_{h}(U_{3}X) is a basis polynomial of the space R(ν, Q_{∈}). Further, it is known ([1], p. 38) that

ϑ(τ, P_{h}(X), Q_{2}) = ϑ(τ, P_{h}(U_{3}X, Q_{2});

thus the theta-series ϑ(τ, P_{h}(X), Q_{2}) and ϑ(τ, P_{h}(U_{3}X), Q_{2}), corresponding to different basis polynomials P_{h}(X) and P_{h}(U_{3}X), are linearly dependent.

Calculate how many such linearly dependent theta-series we have. Let k and i be even (otherwise, it can be shown similarly to (20) that ϑ(τ, P_{h}, Q_{2}) = 0), i.e., k = ν, and i takes \frac{ν}{2} + 1 even values. Thus altogether we have [\frac{1}{2}(\frac{ν}{2} + 1)] linearly dependent theta-series.

But

[\frac{1}{2}(\frac{ν}{2} + 1)] = \begin{cases}\frac{ν}{4} & if ν ≡ 0 (mod 4), \\ \frac{ν+2}{4} & if ν ≡ 2 (mod 4). \end{cases}

Hence

dim T (ν, Q_{2}) ≤ \frac{ν}{2} + 1 − \frac{ν}{4} = \frac{ν}{4} + 1 if ν ≡ 0 (mod 4),

dim T (ν, Q_{2}) ≤ \frac{ν}{2} + 1 − \frac{ν + 2}{4} = \frac{ν + 2}{4} if ν ≡ 2 (mod 4).

(24)

Consider now the quadratic form

Q_{3} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2} + b_{12}x_{1}x_{2},

where 0 < |b_{12}| < b_{11} < b_{33}.

It is easy to verify that the integral automorphs of the quadratic form Q_{3} are

∥\begin{matrix}e_{1} & 0 & 0 \\ 0 & e_{1} & 0 \\ 0 & 0 & e_{2} \end{matrix}∥ and ∥\begin{matrix}0 & e_{1} & 0 \\ e_{1} & 0 & 0 \\ 0 & 0 & e_{2} \end{matrix}∥ (e_{i} = ±1, i = 1, 2).  (25)

From automorphs (25) we use

U_{4} = ∥\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix}∥ and U_{5} = ∥\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}∥ .  (26)

From (9) and (26) it follows that

\begin{matrix}P_{h}(U_{4}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}(−x_{3})^{ν−k} = ∑_{k=0}^{ν} ∑_{i=0}^{k}(−1)^{ν−k}a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k}, \\ P_{h}(U_{5}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}x_{2}^{i}x_{1}^{k−i}x_{3}^{ν−k} = ∑_{k=0}^{ν} ∑_{i=0}^{k} a_{ki}^{(h)}x_{1}^{k−i}x_{2}^{k−(k−i)}x_{3}^{ν−k}. \end{matrix}(27)

We have

P_{h}(X) + P_{h}(U_{4}X) = ∑_{k=0}^{ν} ∑_{i=0}^{k}(1 + (−1)^{k})a_{ki}^{(h)}x_{1}^{i}x_{2}^{k−i}x_{3}^{ν−k}.  (28)

Calculate how many polynomials P_{h} satisfy the equality

P_{h}(X) + P_{h}(U_{4}X) = 0.  (29)

According to (28), we must calculate how many coefficients a_{ki}, with ν − 1 ≤ k ≤ ν and 0 ≤ i ≤ k, have the index k odd, i.e., k = ν − 1. Thus altogether we have ν polynomials P_{h} which satisfy equality (29). But for such polynomials, according to the Lemma 1,

ϑ(τ, P_{h}, Q_{3}) = 0.  (30)

From (14) and (27) it follows that if all last 2ν + 1 coefficients of the basis polynomial P_{h}(X) are equal to zero, except one a_{ki}^{(h)} = 1, then all last 2ν + 1 coefficients of polynomial P_{h}(U_{5}X) are equal to zero, except one a_{k,k−i} = 1.

Hence P_{h}(U_{5}X) is a basis polynomial of the space R(ν, Q_{∋}). Further, it is known ([1], p. 38) that

ϑ(τ, P_{h}(X), Q_{3}) = ϑ(τ, U_{5}(X), Q_{3});

thus the theta-series ϑ(τ, P_{h}(X), Q_{3}) and ϑ(τ, P_{h}(U_{5}X), Q_{3}), corresponding to the different basis polynomials P_{h}(X) and P_{h}(U_{5}X), are linearly dependent.

We count the number of linearly dependent theta-series. Let k be even (otherwise (30) holds true) i.e., k = ν, i takes ν +1 values. Thus we have [\frac{1}{2}(ν +1)] = \frac{ν}{2} linearly dependent theta-series.

We have thus shown that among 2ν +1 theta-series, corresponding to linearly independent spherical polynomials, ν theta-series are zero and ν/2 are linearly dependent theta-series. Hence the maximal number of linearly independent theta-series

dim T (ν, Q_{3}) ≤ 2ν + 1 − ν − \frac{ν}{2} = \frac{ν}{2} + 1.  (31)

Consider the quadratic forms:

Q_{4} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2} + b_{23}x_{2}x_{3},
Q_{5} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2} + b_{13}x_{1}x_{3},
Q_{6} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2} + b_{13}(x_{1}x_{3} + x_{2}x_{3}),
Q_{7} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2} + b_{13}(x_{1}x_{3} + x_{2}x_{3}) + b_{12}x_{1}x_{2},

where 0 < |b_{13}| < b_{11} < b_{33}, |b_{23}| < b_{22}. For each quadratic form Q_{i} (i = 4, 5, 6, 7) we construct the corresponding integral automorphs; consider all possible polynomials P_{h}(U_{j}X) (h = 1, 2, …, 2ν + 1), where P_{h} ∈ R(ν, Q_{⟩}) are spherical basis polynomials of order ν with respect to Q_{i} (see (14)), U_{j} ∈ G are integral automorphs of the quadratic form Q_{i}. We investigate the sets {ϑ(τ, P_{h}(U_{j}), Q_{i}}_{h,j}. From these sets we remove the zero theta-series and leave those through which other theta-series are expressed. Thus we obtain:

dim T (ν, Q_{4}), dim T (ν, Q_{5}), dim T (ν, Q_{6}), dim T (ν, Q_{7}) ≤ ν + 1.

176 K. SHAVGULIDZE

In ([4], p. 31) we have shown that for the quadratic forms Q_{8} = b_{11}x_{1}^{2} + b_{22}(x_{2}^{2} + x_{3}^{2}), Q_{9} = b_{11}x_{1}^{2} + b_{22}x_{2}^{2} + b_{33}x_{3}^{2} + b_{ij}x_{i}x_{j} with 1 ≤ i < j ≤ 3, Q_{10} = b_{11}x_{1}^{2} + b_{22}(x_{2}^{2} + x_{3}^{2}) + b_{1j}x_{1}x_{j} with j = 2, 3, Q_{11} = b_{11}x_{1}^{2} + b_{22}(x_{2}^{2} + x_{3}^{2}) + b_{12}(x_{1}x_{2} + x_{1}x_{3}), Q_{12} = b_{11}x_{1}^{2} + b_{22}(x_{2}^{2} + x_{3}^{2}) + b_{12}(x_{1}x_{2} + x_{1}x_{3}) + b_{23}x_{2}x_{3}, Q_{13} = b_{11}x_{1}^{2} + b_{22}(x_{2}^{2} + x_{3}^{2}) + b_{23}x_{2}x_{3}, dim T (ν, Q_{8}) ≤ \frac{ν}{4} + 1 if ν ≡ 0 (mod 4),

dim T (ν, Q_{8}) ≤ \frac{ν + 2}{4} if ν ≡ 2 (mod 4), dim T (ν, Q_{9}), dim T (ν, Q_{10}), dim T (ν, Q_{11}), dim T (ν, Q_{12}) ≤ ν + 1,

dim T (ν, Q_{13}) ≤ \frac{ν}{2} + 1.

4. Dimension and the Basis of the Space T (ν, Q) for Some

Ternary Forms

From (24), for ν =2 and ν =4, it follows that dim (2, Q_{2})≤1 and dim (4, Q_{2})≤ 2, respectively. Let us show that dim (2, Q_{2}) = 1 and dim (4, Q_{2}) = 2.

For the quadratic form

Q_{2} = b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2},

where 0 < b_{11} < b_{33}, we have D = det A = 8b_{11}^{2}b_{33}, A_{11} = A_{22} = 4b_{11}b_{33}, A_{33} = 4b_{11}^{2}. Hence, according to (5) and (6),

φ_{11} = x_{1}^{2} − \frac{A_{11}}{3D} 2Q_{2} = x_{1}^{2} − \frac{1}{3b_{11}} Q_{2},  (32)
φ_{1111} = x_{1}^{4} − \frac{6}{7b_{11}} x_{1}^{2}Q_{2} + \frac{3}{35b_{11}^{2}} Q_{2}^{2},  (33)
φ_{3333} = x_{3}^{4} − \frac{6}{7b_{33}} x_{3}^{2}Q_{2} + \frac{3}{35b_{33}^{2}} Q_{2}^{2}.  (34)

Consider the equation

b_{11}(x_{1}^{2} + x_{2}^{2}) + b_{33}x_{3}^{2} = n.  (35)

1) When n = b_{11}, equation (35) has 4 solutions:

\begin{matrix}x_{1} = ±1, & x_{2} = x_{3} = 0, \\ x_{2} = ±1, & x_{1} = x_{3} = 0. \end{matrix}

2) When n = b_{33},

a) if b_{33} ≠ b_{11}h^{2} and b_{33} ≠ b_{11}(u^{2} + v^{2}), then (35) has 2 solutions: x_{1} = x_{2} = 0, x_{3} = ±1. b) if b_{33} = b_{11}h^{2} and b_{33} ≠ b_{11}(u^{2} + v^{2}), then (35) has 6 solutions:

\begin{matrix}x_{1} = x_{2} = 0, & x_{3} = ±1; \\ x_{1} = ±h, & x_{2} = x_{3} = 0; \\ x_{2} = ±h, & x_{1} = x_{3} = 0. \end{matrix}

c) if b_{33} ≠ b_{11}h^{2} and b_{33} = b_{11}(u^{2} + v^{2}), then (35) has 10 solutions if u ≠ v

and 6 solutions if u = v:

x_{1} = x_{2} = 0, x_{3} = ±1;
x_{1} = ±u,  x_{2} = ±v, x_{3} = 0;
x_{1} ± v,  x_{2} = ±u, x_{3} = 0;
x_{1} = ±u,  x_{2} = ∓v, x_{3} = 0;
x_{1} = ±v,  x_{2} = ∓u, x_{3} = 0.

d) if b_{33} = b_{11}h^{2} and b_{33} = b_{11}(u^{2} + v^{2}), then (35) has 14 solutions:

x_{1} = x_{2} = 0, x_{3} = ±1;
x_{1} = ±h,  x_{2} = x_{3} = 0;
x_{2} = ±h,  x_{1} = x_{3} = 0;
x_{1} = ±u,  x_{2} = ±v,  x_{3} = 0;
x_{1} = ±v,  x_{2} = ±u,  x_{3} = 0;
x_{1} = ±u,  x_{2} = ∓v,  x_{3} = 0;
x_{1} = ±v,  x_{2} = ∓u,  x_{3} = 0.

Using these solutions and performing easy calculations by (32) we obtain

ϑ(τ, φ_{11}, Q_{2}) = ∑_{n=1}^{∞} ( ∑_{Q_{2}=n} x_{1}^{2} − \frac{1}{3b_{11}} Q_{2}) = z^{n} + ⋯ + \frac{2}{3} z^{b_{11}} + · · · .

Since ϑ(τ, φ_{11}, Q_{2}) ∈ T (2, Q_{2}) and, as is shown above, dim T (2, Q_{2}) ≤ 1, we have proved

Theorem 1. dim T (2, Q_{2}) = 1 and the generalized ternary theta-series ϑ(τ, φ_{11}, Q_{2}) is a basis of the space T (2, Q_{2}).

Performing easy calculations and using the solutions of equation (35), for b_{33} ≠ b_{11}h^{2} and b_{33} ≠ b_{11}(u^{2} + v^{2}) by (33) and (34) we obtain

ϑ(τ, φ_{1111}, Q_{2}) = ∑_{n=1}^{∞} ( ∑_{Q_{2}=n} x_{1}^{4} − \frac{6}{7b_{11}} Q_{2} + \frac{3}{35b_{11}^{2}} Q_{2}^{2})z^{n}

= ⋯ + \frac{22}{35} z^{b_{11}} + ⋯ + \frac{6}{35} \frac{b_{33}^{2}}{b_{11}^{2}} z^{b_{33}} + ⋯,

ϑ(τ, φ_{3333}, Q_{2}) = ∑_{n=1}^{∞} ( ∑_{Q_{2}=n} x_{3}^{4} − \frac{6}{7b_{33}} x_{3}^{2}Q_{2} + \frac{3}{35b_{33}^{2}} Q_{2}^{2})z^{n}

= ⋯ + \frac{12}{35} \frac{b_{11}^{2}}{b_{33}^{2}} z^{b_{11}} + ⋯ + \frac{16}{35} z^{b_{33}} + · · · .

These generalized ternary theta-series are linearly independent since the determinant of second order constructed from the coefficients of these thetaseries is not equal to zero. Similarly, in the remaining cases, the theta-series ϑ(τ, φ_{1111}, Q_{2}) and ϑ(τ, φ_{3333}, Q_{2}) are linearly independent. Above it is shown that dim T (4, Q_{2}) ≤ 2; hence dim T (4, Q_{2}) = 2. Thus we have proved

Theorem 2. dim T (4, Q_{2}) = 2 and the generalized ternary theta-series ϑ(τ, φ_{1111}, Q_{2}) and ϑ(τ, φ_{3333}, Q_{2}) form a basis of the space T (4, Q_{2}).

The proof of the following theorem is similar.

Theorem 3. Let

Q_{1} = b_{11}x_{1}^{2} + b_{22}x_{2}^{2} + b_{33}x_{3}^{2},

then T (2, Q_{1}) = 2 and the generalized ternary theta-series

ϑ(τ, φ_{11}, Q_{2}) = ∑_{n=1}^{∞} ( ∑_{Q_{1}=n} x_{1}^{2} − \frac{1}{3b_{11}} Q_{1})z^{n} = \frac{4}{3} z^{b_{11}} + ⋯ − \frac{2}{3} \frac{b_{22}}{b_{11}} z^{b_{22}} + ⋯,

ϑ(τ, φ_{22}, Q_{2}) = ∑_{n=1}^{∞} ( ∑_{Q_{1}=n} x_{2}^{2} − \frac{1}{3b_{22}} Q_{1})z^{n} = −\frac{2}{3} \frac{b_{11}}{b_{22}} z^{b_{11}} + ⋯ + \frac{4}{3} z^{b_{22}} + ⋯

form a basis of the space T (2, Q_{1}).

References

[1] F. Gooding, Modular forms arising from spherical polynomials and positive definite quadratic forms. J. Number Theory 9(1977), 36–47.

[2] E. Hecke, Mathematische Werke. Zweite Auflage. Vandenhoeck und Ruprecht, Göttingen, 1970.

[3] G. A. Lomadze, On the basis of the space of fourth order spherical functions with respect to a positive quadratic form. (Russian) Bull. Acad. Sci. Georgian SSR 69(1973), No. 3, 533–536.

[4] K. Sh. Shavgulidze, To the dimension of spaces of generalized ternary theta-series. (Russian) Trudy Tbiliss. Univ. Mat. Mekh. Astronom. 26(1989), 22–37.

(Received 11.01.2001)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Georgia E-mail: svanadze@viam.hepi.edu.ge