GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 1997, 385-400

MODULAR PROPERTIES OF THETA-FUNCTIONS AND

REPRESENTATION OF NUMBERS BY POSITIVE QUADRATIC FORMS

T. VEPKHVADZE

Abstract. By means of the theory of modular forms the formulas for a number of representations of positive integers by two positive quaternary quadratic forms of steps 36 and 60 and by all positive diagonal quadratic forms with seven variables of step 8 are obtain.

Let r(n; f) denote a number of representations of a positive integer n by a positive definite quadratic form f with a number of variables s. It is well known that, for the case s > 4, r(n; f) can be represented as

r(n; f) = ρ(n; f) + ν(n; f),

where ρ(n; f) is a “singular series” and ν(n; f) is a Fourier coefficient of cusp form. This can be expressed in terms of the theory of modular forms by stating that

ϑ(τ; f) = E(τ; f) + X(τ),

where E(τ; f) is the Eisenstein series and X(τ) is a cusp form.

In his work [1] Malyshev formulated the following problem: to define the Eisenstein series and to develop a full theory of singular series for arbitrary s ≥ 2. For s ≥ 3, its solution follows from Ramanathan’s results [2].

In the present paper we work out a full solution of this problem. Moreover, convenient formulas are obtained for calculating values of the function ρ(n; f).

Thus, if the genus of the quadratic form f contains one class, then according to Siegel’s theorem ([2], [3], [4]), ϑ(τ; f) = E(τ; f) and in that case the problem for obtaining “exact” formulas for r(n; f) is solved completely. If the genus contains more than one class, then it is necessary to find a cusp form X(τ).

385

1991 Mathematics Subject Classification. 11E20.

Key words and phrases. Quadratic form, number of representations, modular form, generalized theta-function.

1072-947X/97/0700-0385$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

A large number of papers is devoted to finding such formulas. Cusp forms in these works are constructed in the form of linear combinations of products of simple theta-functions with characteristics or their derivatives (see, e.g., [5]), products of Jacobi theta-functions or their derivatives (see, e.g., [6]), and theta-functions with spherical polynomials (see, e.g., [7]). All these functions are certain particular cases of linear combinations of the so-called generalized theta-functions with characteristics defined below by the formula (1). In the present paper, using modular properties of these functions, we have obtained by the unique method the exact formulas for a number of representations of numbers by positive quadratic forms both with an even (forms of such a kind were considered earlier) and an odd number of variables. Moreover, it is shown that using [8], one can reduce cumbersome calculations for obtaining formulas for a number of representations of numbers by the quadratic forms considered earlier and to obtain new formulas.

§ 1. Let f = \frac{1}{2}x′Ax be a positive definite qadratic form, let A be an integral matrix with even diagonal elements, and the vector column x ∈ ℤ^{s}, s ∈ N, s ≥ 2. We call u ∈ ℤ^{s} a special vector with respect to the form f if Au ≡ 0 (mod N), where N is a step of the form f. Moreover, let P_{ν} = P_{ν}(x) be a spherical function of the νth order (ν is a positive integer) corresponding to the form f (see [9], p. 454). Then the generalized thetafunction with characteristics we define as follows:

ϑ_{gh}(τ; p_{ν}, f) ∑_{x≡g (mod N)}(−1)^{\frac{h′A(x−g)}{N^{2}}} p_{ν}(x)e^{\frac{πiτx′Ax}{N^{2}}} ;  (1)

here and below, g and h are the special vectors with respect to the form f.

In the sequel, use will be made of the following lemmas (see Lemmas 1 and 4 in [8]).

Lemma 1. Let k be an arbitrary integral vector and l be a special vector with respect to the form f. Then the following equalities hold:

ϑ_{g+Nk,h}(τ; p_{ν}, f) = (−1)^{\frac{h′Ak}{N}} ϑ_{gh}(τ; p_{ν}, f),
ϑ_{g,h+2l}(τ; p_{ν}, f) = ϑ_{gh}(τ; p_{ν}, f).

Lemma 2. Let F (τ) be an entire modular form of the type (−r, N, v(L)) and let there exist an integer l for which (v(L))^{l} = 1. Then the function F (τ) is identically equal to zero if in its expansion in powers of Q = e^{2πiτ} the coefficients c_{n} equal 0 for n ≤ (r/12)N ∏_{p|N}(1 + p^{−1}).

In the main theorem below we formulate modular properties of linear combinations of functions (1).

Theorem 1. Let f_{1} = f_{1}(x) = \frac{1}{2}x′A_{1}x, …, f_{j} = f_{j}(x) = \frac{1}{2}x′A_{j}x be positive definite quadratic forms with a number of variables s, let P_{ν}^{(k)} = P_{ν}^{(k)}(x) be the coresponding spherical functions of order ν, Δ_{k} be the determinant of the matrix A_{k}, N_{k} the step of the form f_{k} (k = 1, …, j), Δ the determinant of the matrix A of some positive definite quadratic form \frac{1}{2}x′Ax with a number of variables s + 2ν and the step N.

Next, let g^{(k)} and h^{(k)} be special vectors with respect to f_{k}; moreover, given 2 ∤ \frac{N}{N_{k}}, let h_{k} be the vector with even components (k = 1, …, j);

L = (\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈ Γ_{0}(N);

v(L) = (i^{\frac{1}{2} η(γ)(sgn δ−1)})^{s+2ν}(i^{(\frac{|δ|−1}{2})^{2}})^{s+2ν}(\frac{2(sgn δ)βΔ}{|δ|}) for 2 ∤ s,
= (sgn δ)^{\frac{s}{2} +ν}(\frac{(−1)^{\frac{s}{2} +ν}Δ}{|δ|}) for 2 | s;  (2)

η(γ) = 1 for γ ≥ 0,
= −1 for γ < 0;

(\frac{(−1)^{\frac{s}{2} +ν}Δ}{|Δ|}) is the Kronecker symbol, (\frac{2(sgn δ)βΔ}{|δ|}) is the Jacobi symbol.

Then the function

X(τ) = ∑_{k=1}^{j} B_{k}ϑ_{g^{(k)}h^{(k)}}(τ; P_{ν}^{(k)}, f_{k})  (3)

for arbitrary complex numbers B_{k} is an entire modular form of the type (−(\frac{s}{2} + ν), N, v(L)) if and only if the conditions

N_{k}|N, N_{k}^{2}|f_{k}(g^{(k)}), 4N_{k}|\frac{N}{N_{k}} f_{k}(h^{(k)})

are fulfilled, and for all α and δ such that αδ ≡ 1 (mod N) we have

∑_{k=1}^{j} B_{k}ϑ_{αg^{(k)},h^{(k)}}(τ; p_{ν}^{(k)}, f_{k})(sgn δ)^{ν}(\frac{(−1)^{[ \frac{1}{2} ]}Δ_{k}}{|δ|} ) =

= (\frac{(−1)^{[ \frac{s+2ν}{2} ]}Δ}{|δ|}) ∑_{k=1}^{j} B_{k}ϑ_{g^{(k)}h^{(k)}}(τ; p_{ν}^{(k)}, f_{k}).

This theorem has been proved in [8] for even h_{k}. It can easily be adjusted to the case 2|\frac{N}{N_{k}} h_{k}. From this theorem we obtain the following two theorems which are analogues of Theorems 4 and 2 from [8].

Theorem 2. If all the conditions of Theorem 1 are fulfilled and either ν > 0, or ν = 0 and all the g^{(k)} vectors are nonzero, then the function (3) is a cusp form of the type (−(\frac{1}{2} + ν), N, v(L)).

Theorem 3. Let f be an integral positive quadratic form with a number of variables s and let Δ be a determinant of the form f. Then the function ϑ(τ; f) defined by the formula

ϑ(τ; f) = 1 + ∑_{n=1}^{∞} r(n; f)e^{2πiτf} (Im τ > 0)  (4)

is the entire modular form of the type (−\frac{s}{2}, N, ν(L)), where ν(L) are defined by the formulas (2) for ν = 0.

From the results of [2], [3], [4] and [10] we obtain

Theorem 4. Let f be a positive quadratic form with a number of variables s and let Δ be its determinant. Then the function E(τ, z; f), determined for Re z ≥ 2 − \frac{s}{2} and Im τ > 0 by the formula

E(τ, z; f) = 1 + \frac{e^{\frac{πis}{4}}}{2^{\frac{s}{2}} Δ^{\frac{s}{2}}} ∑_{q=1}^{∞} ∑_{\begin{matrix}H=−∞ \\ (H,q)=1 \end{matrix}}^{∞} \frac{S(fh, q)}{q^{\frac{s}{2}} (qτ − H)^{\frac{s}{2}} |qτ − H|^{z}} ,

where S(fh, q) is the Gaussian sum, can be continued analytically into the neighborhood of the point z = 0. Further, having defined the Eisenstein series E(τ; f) by the formulas

E(τ; f) = \frac{1}{2}E(τ, z; f)|_{z=0} for s = 2
= E(τ, z; f)|_{z=0} for s > 2,

we have

E(τ; f) = 1 + \frac{1}{2} ∑_{n=1}^{∞} ρ(n; f)e^{2πiτn} for s = 2,

= 1 + ∑_{n=1}^{∞} ρ(n; f)e^{2πiτn} for s > 2;  (5)

here ρ(n; f) is a singular series which is calculated as follows:

(1) If 2 | s, v = ∏\begin{matrix}p|n \\ p∤2Δ \end{matrix} p^{w}, Δ = r^{2}ω (ω is a square-free number), then

ρ(n; f) = \frac{π^{\frac{s}{2}}}{Γ(\frac{s}{2})Δ^{1/2}} n^{\frac{s}{2} −1}χ_{2} ∏_{\begin{matrix}p|Δ \\ p>2 \end{matrix}} χ_{p} ∏_{\begin{matrix}p|r \\ p>2 \end{matrix}} (1 − (\frac{(−1)^{\frac{s}{2}} ω}{p} )p^{− \frac{s}{2}} )^{−1} ×

× L^{−1}(\frac{s}{2}; (−1)^{\frac{1}{2}} ω) ∑_{k|v} (\frac{(−1)^{\frac{s}{2}} Δ}{k})k^{1− \frac{s}{2}} .

(2) If 2 ∤ s, Δn = 2^{α+γ}v_{1}v_{2} = r^{2}ω, 2^{α}∥n, 2^{γ}∥Δ, p^{l}∥Δ, p^{w}∥n (p > 2),

v_{1} = ∏\begin{matrix}p|n \\ p∤2Δ \end{matrix} p^{w} = r_{1}^{2}ω_{1}, v_{2} = ∏\begin{matrix}p|Δn \\ p|Δ, p>2 \end{matrix} p^{w+l} = r_{2}^{2}ω_{2} (ω, ω_{1}, ω_{2} are square-

free numbers), then

ρ(n; f) = \frac{(s − 1)! r_{1}^{2−s}n^{\frac{s}{2} −1}}{Γ(\frac{s}{2})2^{s−2}π^{\frac{s}{2} −1}|B_{s−1}|Δ^{\frac{1}{2}}} χ_{2} ∏_{\begin{matrix}p|Δ \\ p>2 \end{matrix}} χ_{p} ×

× ∏_{p|2Δ}(1 − p^{1−s})^{−1}L(\frac{s − 1}{2}; (−1)^{\frac{s−1}{2}} ω) ∏_{\begin{matrix}p|r_{2} \\ p>2 \end{matrix}} (1 − (\frac{(−1)^{\frac{s−1}{2}} ω}{p} )p^{\frac{1−s}{2}} ) ×

× ∑_{k|r_{1}} k^{s−2} ∏_{p|k} (1 − (\frac{(−1)^{\frac{s−1}{2}} ω}{p} )p^{\frac{1−s}{2}} ).

The values of χ_{2} and χ_{p} are given in [11] (formulas (9)–(13), p. 66);

L(k; (−1)^{k}ω) = ∑_{\begin{matrix}l=1 \\ 2∤l \end{matrix}}^{∞} (\frac{(−1)^{k}ω}{l} )\frac{1}{l^{k}} = ∏_{\begin{matrix}p \\ p>2 \end{matrix}} (1 − (\frac{(−1)^{k}ω}{p} )p^{−k})^{−1};

B_{s−1} are Bernoulli’s numbers.

§ 2. In this section we will obtain exact formulas for a number of representations of numbers by quaternary quadratic forms:

x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 15x_{4}^{2} and 2x_{1}^{2} + 2x_{1}x_{2} + 5x_{2}^{2} + 2x_{3}^{2} + 2x_{3}x_{4} + 5x_{3}^{2}.

The first form has been considered by Lomadze in [5] who, for the construction of cusp form X(τ), used products of simple theta-functions with characteristics and of their derivatives (some particular cases of the function (1)) and therefore he had to use modular forms of step 240 instead of 60.

Theorem 5. Let

f = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 15x_{4}^{2}, f_{1} = 3x_{1}^{2} + 15x_{2}^{2},

f_{2} = 4x_{1}^{2} + 2x_{1}x_{2} + 4x_{2}^{2}, g^{(1)} = (\begin{matrix}20 \\ 20 \end{matrix}) , h^{(1)} = (\begin{matrix}0 \\ 0 \end{matrix}) ,

g^{(2)} = (\begin{matrix}15 \\ 0 \end{matrix}) , h^{(2)} = (\begin{matrix}0 \\ 15 \end{matrix}) , p_{1}^{(1)} = x_{2}, p_{1}^{(2)} = x_{1} + x_{2}.

Then the equality

ϑ(τ; f) = E(τ; f) + \frac{4}{15}ϑ_{g^{(1)}h^{(1)}}(τ; p^{(1)}, f_{1}) + \frac{1}{10}ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2}) (6)

holds, where the functions

ϑ(τ; f), ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}), ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2})

are defined by formulas (4) and (1), while the function E(τ; f) by formula (5).

Proof. By Theorem 3, the function ϑ(τ; f) belongs to the space of entire modular forms of the type (−2, 60, v(L)), where v(L) is the corresponding multiplier system. Then, according to Siegel’s theorem (see [2]), E(τ; f) also belongs to this space. Using Lemma 1, we can check that the function

X(τ) = \frac{4}{15}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) + \frac{1}{10}ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2})

satisfies all the conditions of Theorem 1.

Indeed, f_{1} is the binary form of step 60 and f_{2} is the binary form of step

30 (N_{1} = 60, N_{2} = 30), g^{(1)} = (\begin{matrix}20 \\ 20 \end{matrix}), and h^{(1)} = (\begin{matrix}0 \\ 0 \end{matrix}) are special vectors

with respect to the form f_{1} = 3x_{1}^{2}+15x_{2}^{2}, and g^{(2)} = (\begin{matrix}15 \\ 0 \end{matrix}) and h^{(2)} = (\begin{matrix}0 \\ 15 \end{matrix})

are special vectors with respect to the form f_{2} = 4x_{1}^{2} +2x_{1}x_{2} +4x_{2}^{2}. 2 | h^{(1)}, 2 | \frac{N}{N_{2}} h^{(2)}, since N_{1} = N = 60, N_{2} = 30; but 60 | N, 30 | N, 60^{2} | f_{1}(g^{(1)}), 30^{2} | f_{2}(g^{(2)}), 240 | f_{1}(h^{(n)}), 120 | 2f_{2}(h^{(2)}).

If αδ ≡ 1 (mod 60), then either

α ≡ δ ≡ 1 (mod 3) or α ≡ δ ≡ −1 (mod 3).

Because of Lemma 1,

ϑ_{αg^{(1)},h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) = \begin{cases}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) for α ≡ δ ≡ 1 (mod 3), \\ ϑ_{−g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) for α ≡ δ ≡ −1 (mod 3). \end{cases}

Due to (1) we have

ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}, f_{1}) = ∑_{x≡g^{(1)} (mod 60)} x_{2}e^{2πiτ \frac{3x^{2}_{1}+15x^{2}_{2}}{60^{2}}}

= − ∑_{x≡−g^{(1)} (mod 60)} x_{2}e^{2πiτ \frac{3x^{2}_{1}+15x^{2}_{2}}{60^{2}}} = −ϑ_{−g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}).

Thus

ϑ_{αg^{(1)},h^{(1)}}(τ; p_{1}^{(1)}, f_{1})=\begin{cases}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) for α≡δ ≡1 (mod 3), \\ ϑ_{−g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) for α≡δ ≡−1 (mod 3). \end{cases} (7)

We have

sgn δ(\frac{−Δ_{1}}{|δ|} ) = sgn δ(\frac{−1}{|δ|})(\frac{5}{|δ|}), (\frac{(−1)^{2}Δ}{|δ|}) = (\frac{−1}{|δ|})(\frac{|δ|}{3})(\frac{5}{|δ|}). (8)

Furthermore, we have

(\frac{|δ|}{3}) = \begin{cases}sgn δ for δ ≡ 1 (mod 3), \\ − sgn δ for δ ≡ −1 (mod 3). \end{cases}

We can easily verify that formulas (7) and (8) imply

ϑ_{αg^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) sgn δ(\frac{−Δ_{1}}{|δ|} ) = (\frac{Δ}{|δ|})ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}^{(1)}). (9)

Analogously, we get

ϑ_{αg^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2}) sgn δ(\frac{−Δ_{2}}{|δ|} ) = (\frac{Δ}{|δ|})ϑ_{αg^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2}). (10)

Consequently, according to (9) and (10), the function

X(τ) = \frac{4}{15}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) + \frac{1}{10}ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2})  (11)

satisfies the conditions of Theorem 1 and, due to Theorem 2, belongs to the space of cusp forms of the type (−2, 60, v(L)).

Thus, owing to Lemma 2, the function

Ψ(τ) = ϑ(τ; f) − E(τ; f) − X(τ),

where X(τ) is defined by (11), will be identically zero if all coefficients for Q^{n}(n ≤ 24) are zero in its expansion in powers of Q = e^{2πiτ}.

Next, let n = 2^{α}3^{β_{1}}5^{β_{2}}u, (u, 30) = 1. Then by Theorem 4,

E(τ; f) = 1 + ∑_{n=1}^{∞} ρ(n; f)Q^{n} (Q = e^{2πiτ} ),  (12)

where

ρ(n; f) = \frac{1}{12}(2^{α+1} + (−1)^{β_{1}}(\frac{−1}{u}))(3^{β_{1}+1} − (−1)^{α+β_{2}}(\frac{u}{3})) ×
× (5^{β_{2}+1} + (−1)^{α+β_{1}+β_{2}}(\frac{u}{5})) ∑_{d_{1}d_{2}=u} (\frac{15}{d_{1}})d_{2}.  (13)

Having calculated the values ρ(n; f) for all n ≤ 24 by formula (13), we obtain because of (12):

E(τ; f) = 1 + 3Q + \frac{20}{3}Q^{2} + \frac{8}{3}Q^{3} + 9Q^{4} + 24Q^{5} + 15Q^{6} +

+ \frac{16}{3}Q^{7} + \frac{68}{3}Q^{8} + 39Q^{9} + \frac{65}{3}Q^{10} + 24Q^{11} + \frac{56}{3}Q^{12} + 24Q^{13} +

+ 48Q^{14} + \frac{65}{3}Q^{15} + 33Q^{16} + 72Q^{17} + \frac{140}{3}Q^{18} + 18Q^{19} +

+ 72Q^{20} + 96Q^{21} + 24Q^{22} + \frac{88}{3}Q^{23} + 75Q^{24} + ….  (14)

Formulas (4) and (1) yield

ϑ(τ; f) = 1 + 6Q + 12Q^{2} + 8Q^{3} + 6Q^{4} + 24Q^{5} + 24Q^{6} +
+ 12Q^{8} + 30Q^{9} + 24Q^{10} + 24Q^{11} + 8Q^{12} + 24Q^{13} +
+ 480Q^{14} + 2Q^{15} + 18Q^{16} + 72Q^{17} + 52Q^{18} + 36Q^{19} +
+ 72Q^{20} + 96Q^{21} + 24Q^{22} + 24Q^{23} + 84Q^{24} + ….  (15)
\frac{4}{15}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) = \frac{16}{3}(Q^{2} + Q^{3} − Q^{7} − 2Q^{8} + Q^{10} −
− 2Q^{12} − 2Q^{15} + Q^{18} − Q^{23} + 4Q^{27} + …),  (16)
\frac{1}{10}ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2}) = \frac{3}{2}(2Q − 2Q^{4} − 6Q^{6} − 6Q^{9} −
− 2Q^{10} − 6Q^{15} − 10Q^{16} + 12Q^{19} + 6Q^{24} + …).  (17)

Taking into account (14)–(17), we can easily verify that all coefficients for Q^{n}(n ≤ 24) in the expansion of the function ψ(τ) in powers of Q are zero. Thus identity (6) is proved.

Theorem 6. Let f = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 15x_{4}^{2}, n = 2^{α}3^{β_{1}}5^{β_{2}}u, (u, 30) = 1. Then

r(n; f) = \frac{1}{6}(3^{β_{1}+1} − (−1)^{β_{2}}(\frac{u}{3})) ×

× (5^{β_{2}+1} + (−1)^{β_{1}+β_{2}}(\frac{u}{5})) ∑_{d_{1}d_{2}=u} (\frac{15}{d_{1}})d_{2} +

+ \frac{3}{2} ∑_{\begin{matrix}n=x_{1}^{2}+x_{1}x_{2}+4x_{2}^{2} \\ 2∤x_{1} \end{matrix}}(−1)^{\frac{x_{1}−1}{2}} (x_{1} + 2x_{2}) for n ≡ 1 (mod 4),

= \frac{1}{12}(2^{α+1} + (−1)^{β_{1}}(\frac{−1}{u}))(3^{β_{1}+1} − (−1)^{α+β_{2}}(\frac{u}{3})) ×

× (5^{β_{2}+1} + (−1)^{α+β_{1}+β_{2}}(\frac{u}{5})) ∑_{d_{1}d_{2}=u} (\frac{15}{d_{1}})d_{2} +

+ \frac{3}{2} ∑_{\begin{matrix}n=x_{1}^{2}+x_{1}x_{2}+4x_{2}^{2} \\ 2∤x_{1} \end{matrix}}(−1)^{\frac{x_{1}−1}{2}} (x_{1} + 2x_{2}) +

+ \frac{16}{3} ∑_{\begin{matrix}3n=x_{1}^{2}+5x_{2}^{2} \\ x_{1}≡x_{2}≡1 (mod 3) \end{matrix}} x_{2} otherwise.

Proof. Equating coefficients of the same powers Q in both parts of identity (6), we get

r(n; f) = ρ(n; f) + \frac{16}{3}ν_{1}(n) + \frac{3}{2}ν_{2}(n),  (18)

where ν_{1}(n), ν_{2}(n) denote respectively the coefficients for Q in the expansions of the functions

\frac{1}{20}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}), \frac{1}{15}ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2})

in powers of Q.

From (1) we have

\frac{1}{20}ϑ_{g^{(1)}h^{(1)}}(τ; p_{1}^{(1)}, f_{1}) = ∑_{x_{1},x_{2}=−∞}(3x_{2} + 1)e^{\frac{2πiτ((3x_{1}+1)^{2}+5(3x_{2}+1)^{2})}{3}} ,

i.e.,

ν_{1}(n) = ∑_{\begin{matrix}3n=x_{1}^{2}+5x_{2}^{2} \\ x_{1}≡x_{2}≡1 (mod 3) \end{matrix}} x_{2}.  (19)

It follows from (4) that

\frac{1}{15}ϑ_{g^{(2)}h^{(2)}}(τ; p_{1}^{(2)}, f_{2}) =

= ∑_{x_{1},x_{2}=−∞}(−1)^{x_{1}}(2x_{1} + 1 + 2x_{2})e^{\frac{2πiτ[(2x_{1}+1)^{2}+(2x_{1}+1)x_{2}+4x^{2}_{2}]}{1}} ,

i.e.,

ν_{2}(n) = ∑_{\begin{matrix}n=x_{1}^{2}+x_{1}x_{2}+4x_{2}^{2} \\ 2∤x_{1} \end{matrix}}(−1)^{\frac{x_{1}−1}{2}} (x_{1} + 2x_{2}).  (20)

From formulas (18), (13), (19), (20) we obtain the desired exspression for r(n; f).

Theorem 7. Let f = 2x_{1}^{2} + 2x_{1}x_{2} + 5x_{2}^{2} + 2x_{3}^{2} + 2x_{3}x_{4} + 5x_{4}^{2}, f_{1} =

3x_{1}^{2} + 9x_{2}^{2}, g = (\begin{matrix}18 \\ 6 \end{matrix}), h = (\begin{matrix}0 \\ 0 \end{matrix}), p_{1} = x_{2}. Then

ϑ(τ; f) = E(τ; f) − \frac{1}{9}ϑ_{gh}(τ; p_{1}, f_{1}),

where the functions ϑ(τ; f), E(τ; f) and ϑ_{gh}(τ; p_{1}, f_{1}) are defined respectively by the formulas (4), (5) and (1).

Proof. Let n = 2^{α}3^{β}u, (u, 6) = 1. Then by Theorem 4, E(τ; f) = 1 + ∑_{n=1}^{∞} ρ(n; f)Q^{n} (Q = e^{2πiτ} ), where

ρ(n; f) = 12(3^{β−1} − 1) ∑_{μ|u} μ for α > 0, β > 0,

= 4(3^{β−1} − 1) ∑_{μ|u} μ for α = 0, β > 0,

= 4 ∑_{μ|u} μ for α > 0, β > 0,

= \frac{4}{3} ∑_{μ|u} μ for (n, 6) = 1.  (21)

Formulas (5) and (21) imply

\begin{matrix}E(τ; f) = 1 + \frac{4}{3}Q + …, ϑ(τ; f) = 1 + 4Q^{2} + …, \\ −\frac{1}{9}ϑ_{9h}(τ; p_{1}, f_{1}) = −\frac{4}{3}Q + …. \end{matrix}  (22)

By Theorem 3, the function ϑ(r; f) belongs to the space of entire modular forms of the type (−2, 36, 1). Then by Siegel’s theorem (see [2]), E(τ; f) also belongs to this space. Using Lemma 1, we can easily verify that the function ϑ_{gh}(τ; p_{1}, f_{1}) satisfies all the conditions of Theorem 1. Therefore by Theorem 2, it belongs to the space of cusp forms of the type (−2, 36, 1). It is well known that this space is one-dimensional (see [12]). Therefore from (22) we obtain the above assertion.

From Theorem 7 immediately follows

Theorem 8. Let n = 2^{α}3^{β}u, (u, 6) = 1, f = 2x_{1}^{2} + 2x_{1}x_{2} + 5x_{2}^{2} + 2x_{3}^{2} + 2x_{3}x_{4} + 5x_{4}^{2}. Then

r(n; f) = 12(3^{β−1} − 1) ∑_{μ|u} μ for α > 0, β > 0,

= 4(3^{β−1} − 1) ∑_{μ|u} μ for α = 0, β > 0,

= 4 ∑_{μ|u} μ for α > 0, β = 0,

= \frac{4}{3} ∑_{μ|u} μ − \frac{2}{3} ∑_{\begin{matrix}4n=3x_{1}^{2}+x_{2}^{2} \\ x_{1}≡1 (mod 2) \\ x_{2}≡1 (mod 6) \end{matrix}} x_{2} for (n, 6) = 1.

Remark to Theorem 8. Let

ν(n) = \frac{1}{2} ∑_{\begin{matrix}4n=3x_{1}^{2}+x_{2}^{2} \\ x_{1}≡1 (mod 2) \\ x_{2}≡1 (mod 6) \end{matrix}} x_{2}.

It can be easily shown that

ν(n) = \frac{1}{2} ∑_{\begin{matrix}4n=3x_{1}^{2}+x_{2}^{2} \\ x_{1}≡1 (mod 2) \\ x_{2}≡1 (mod 6) \end{matrix}}(x_{1} + x_{2}).

Further, arguing as in [12] (p. 233), we can easily show that

(1) ν(n_{1}n_{2}) = ν(n_{1})ν(n_{2}) if (n_{1}, n_{2}) = 1;

(2) ν(p^{β}) = ∑_{0≤k< \frac{β}{2}} p^{k}T r(π^{β−2k}(p)) + δ(\frac{β}{2} )p^{\frac{β}{2}} ,

where π(p) is the Frobenius endomorphism of a curve y^{2} = x^{3} +1 reduced in modulo p, δ(r) is equal to one or to zero according to whether the number r is an integer or not. In particular, if n = p is a prime number, then

ν(p) = − ∑_{x=0}^{p−1} (\frac{x^{3} + 1}{p}),

where (\frac{x^{3}+1}{p}) is the Legendre symbol.

§ 3. In this section we obtain formulas for a number of representations of numbers by quadratic forms with seven variables

f = 2 ∑_{j=1}^{s} x_{j}^{2} + ∑_{j=s+1}^{7} x_{j}^{2} (1 ≤ s ≤ 6).  (23)

The cases s = 0 and s = 1 are considered earlier (see, e.g., [13], Vol. II, pp. 305, 309, 335 and Vol. III, p. 237). In these cases the corresponding forms belong to one-class genera. The case s = 3 was considered in [6].

Theorem 9. Let f be of the kind (23), f_{1} = 2x_{1}^{2} + 2x_{2}^{2} + x_{3}^{2}, f_{2} = x_{1}^{2} +

x_{2}^{2} + 2x_{3}^{2}, g^{(1)} = (\begin{matrix}4 \\ 4 \\ 0 \end{matrix}), h^{(1)} = (\begin{matrix}2 \\ 2 \\ 4 \end{matrix}), p_{2} = x_{1}x_{2}, g^{(2)} = (\begin{matrix}4 \\ 4 \\ 4 \end{matrix}), h^{(2)} = (\begin{matrix}4 \\ 4 \\ 0 \end{matrix}).

Then the following equality holds,

ϑ(τ; f) = E(τ; f) + X(τ; f),  (24)

where

X(τ; f) = \frac{1}{32}ϑ_{g^{(1)}h^{(1)}}(τ; p_{2}, f_{1}) for s = 2, 4,

= \frac{1}{32(s − 1)}ϑ_{g^{(2)}h^{(2)}}(τ; p_{2}, f_{2}) for s = 3, 5,

= 0 otherwise.  (25)

Proof. By Theorem 3, the functions ϑ(τ; f) belong to two different spaces of modular forms (−\frac{7}{2}, 8, v(L)), where v(L) is a system of multipliers corresponding to the form f. This system is the same for all s with the same evenness. Then, according to Siegel’s theorem (see [1]), the functions E(τ; f) also belong to appropriate spaces of modular forms. It can be easily verified that functions (25) satisfy all the conditions of Theorem 1 and, by Theorem 2, they belong to two different spaces of cusp forms depending on s.

Let n = 2^{α}u (2 ∤ u, α ≥ 0), 2^{s}n = r_{s}^{2}ω_{s}, u = r^{2}ω (s = 1, 2, …, 6) and let ω and ω_{s} be square-free numbers. Then by Theorem 4 we have

E(τ; f) = 1 + ∑_{n=1}^{∞} ρ(n; f)Q^{n} (Q = e^{2πiτ}),  (26)

where

ρ(n; f) = 2^{\frac{5α}{2} +9− \frac{s}{2}} π^{−3}ω^{\frac{5}{2}} L(3; −ω_{s})χ_{2} ∑_{μ|r} μ^{5} ∏_{p|μ} (1 − (\frac{−ω_{s}}{p} )p^{−3}).

MODULAR PROPERTIES OF THETA-FUNCTIONS 397

By Lemma 27 from [14] we have

L(3; −1) = \frac{π^{3}}{32} , L(3; −2) = \frac{3π^{3}}{64\sqrt{2}};  (27)
L(3; −ω) = \frac{π^{3}}{16ω^{\frac{5}{2}}} \{ ∑_{1≤h≤ \frac{ω}{4}} (ω^{2} − 16h^{2})(\frac{h}{ω} ) + 3ω^{2} ∑_{\frac{ω}{4} <h< \frac{ω}{2}} (\frac{h}{ω} ) +

+ 16 ∑_{\frac{ω}{4} <h≤ \frac{ω}{2}} h(h − ω)(\frac{h}{ω} )\}, if ω ≡ 1 (mod 4), ω > 1,

= \frac{π^{3}}{2ω^{\frac{5}{2}}} ∑_{1≤h≤ \frac{ω}{2}} h(ω − 2h)(\frac{h}{ω} ), if ω ≡ 3 (mod 4),

= \frac{π^{3}}{32ω^{\frac{5}{2}}} \{ ∑_{1≤h≤ \frac{ω}{16}}(3ω^{2} − 256h^{2})( \frac{h}{\frac{1}{2}ω} ) +

+ 4ω ∑_{\frac{ω}{16} <h< \frac{3ω}{16}} (ω − 8h)( \frac{h}{\frac{1}{2}ω} ) + 13ω^{2} ∑_{\frac{3ω}{16} <h≤ \frac{ω}{4}} ( \frac{h}{\frac{1}{2}ω} ) −

− 128 ∑_{\frac{3ω}{16} <h≤ \frac{ω}{4}} h(ω − 2h)( \frac{h}{\frac{1}{2}ω} )\}, if ω ≡ 2 (mod 8), ω > 2,

= \frac{π^{3}}{32ω^{\frac{5}{2}}} \{32ω ∑_{1≤h≤ \frac{ω}{16}} h( \frac{h}{\frac{1}{2}ω} ) − ω^{2} ∑_{\frac{ω}{16} <h< \frac{3ω}{16}} ( \frac{h}{\frac{1}{2}ω} ) +

+ 64 ∑_{\frac{ω}{16} <h≤ \frac{3ω}{16}} h(ω − 4h)( \frac{h}{\frac{1}{2}ω} ) + 8ω ∑_{\frac{3ω}{16} <h≤ \frac{ω}{4}} (ω − 4h)( \frac{h}{\frac{1}{2}ω} )\},

if ω ≡ 6 (mod 8).  (28)

Using formulas (33) of [9], after calculation of values χ_{2}, we obtain

χ_{2} = 1 for 2 ∤ s, α = 0, or for 2|s, α = 0, u ≡ 1 (mod 4),
or 2|s, α = 1,

= 1 + (−1)^{\frac{u^{2}−1}{6}} 2^{\frac{s}{2} −5}, for 2|s, α = 0, u ≡ 3 (mod 4),

= 1 + \frac{2^{\frac{s}{2} −3}(1 − 2^{− \frac{5α}{2}} · 63)}{31} for 2|s, 2|α, u ≡ 1 (mod 4),

= 1 + \frac{2^{\frac{s}{2} −3}(1 − 2^{− \frac{5α}{2}} + (−1)^{\frac{u^{2}−1}{8}} 2^{− \frac{5α}{2} −2} · 31)}{31} for 2|s,

2|α, u ≡ 3 (mod 4)

= 1 + \frac{2^{\frac{s}{2} −3}(1 − 2^{− \frac{5α}{2} + \frac{5}{2}} · 63)}{31} for 2|s, 2 ∤ α, α > 1,

= 1 + \frac{2^{\frac{s}{2} − \frac{1}{2}} (1 − 2^{− \frac{5α}{2}} · 63)}{31} for 2 ∤ s, 2|α, α > 0,

= 1 + \frac{2^{\frac{s}{2} − \frac{1}{2}} (1 − 2^{− \frac{5α}{2} − \frac{5}{2}} · 63)}{31} for 2 ∤ s, 2 ∤ α, u ≡ 1 (mod 4),

= 1 + \frac{2^{\frac{s}{2} − \frac{1}{2}} (1 − 2^{− \frac{5α}{2} − \frac{5}{2}} + (−1)^{\frac{u^{2}−1}{8}} 2^{− \frac{5α}{2} − \frac{9}{2}} · 31)}{31}

for 2 ∤ s, 2 ∤ α, u ≡ 3 (mod 4).  (29)

By (1) we have

ϑ_{g^{(1)}h^{(1)}}(τ; p_{2}, f_{1}) = 16 ∑_{x_{1},x_{2},x_{3}=−∞}^{∞}(−1)^{x_{1}+x_{2}+x_{3}}(2x_{1} + 1)(2x_{2} + 1) ×

× e^{\frac{2πiτ[(2x_{1}+1)^{2}+(2x_{2}+1)^{2}+2x^{2}_{3}]}{2}} ,  (30)
ϑ_{g^{(2)}h^{(2)}}(τ; p_{2}, f_{2}) = 16 ∑_{x_{1},x_{2},x_{3}=−∞}^{∞}(−1)^{x_{1}+x_{2}}(2x_{1} + 1)(2x_{2} + 1) ×

× e^{\frac{2πiτ[(2x_{1}+1)^{2}+(2x_{2}+1)^{2}+2(2x_{3}+1)^{2}]}{4}} .  (31)

Taking then into account (26)–(31) and arguing as in the proof Theorem 5, we obtain the above assertion.

From Theorem 9 we have

Theorem 10. Let n = 2^{α}u (2 ∤ u, α ≥ 0), 2^{s}n = r_{1}^{2}ω_{s}, u = r^{2}ω, and let ω and ω_{s} be square-free numbers, s = 1, 2, …, 6,

f = 2 ∑_{j=1}^{s} x_{j}^{2} + ∑_{j=s+1}^{7} x_{j}^{2}.

Then

r(n; f) = 2^{\frac{5α}{2} − \frac{s}{2} +9}ω^{\frac{5}{2}} π^{−3}L(3; −ω_{s})χ_{2} ×

× ∑_{μ|r} μ^{5} ∏_{p|μ} (1 − (\frac{−ω_{s}}{p} )p^{−3}) + ν(n; f),

where

ν(n; f) = 0 for s = 1, 6,

= \frac{1}{2} ∑_{\begin{matrix}2n=x_{1}^{2}+x_{2}^{2}+2x_{3}^{2} \\ 2∤x_{1}, 2∤x_{2} \end{matrix}}(−1)^{\frac{x_{1}x_{2}−1}{2} +x_{3}}x_{1}x_{2} for s = 2, 4,

= \frac{1}{2s − 2} ∑_{\begin{matrix}4n=x_{1}^{2}+x_{2}^{2}+2x_{3}^{2} \\ 2∤x_{1}, 2∤x_{2}, 2∤x_{3} \end{matrix}}(−1)^{\frac{x_{1}x_{2}−1}{2}} x_{1}x_{2} for s = 3, 5.

The values L(3; −ω_{3}) and χ_{2} can be calculated by formulas (27)–(29).

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(Received 22.01.1995)

Author’s address: Faculty of Physics I. Javakhishvili Tbilisi State University 1, I. Chavchavadze Ave., Tbilisi 380028 Georgia