GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 5, 1996, 485-500

ON SOME ENTIRE MODULAR FORMS OF WEIGHTS \frac{7}{2} AND \frac{9}{2} FOR THE CONGRUENCE GROUP Γ_{0}(4N)

G. LOMADZE

Abstract. Entire modular forms of weights \frac{7}{2} and \frac{9}{2} for the congruence group Γ_{0}(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 7 and 9 variables.

1.

In this paper N, a, k, q, r, t denote positive integers; u, s are odd positive integers; H, c, g, h, j, m, n, α, β, γ, δ, ξ are integers; A, B, C, D, G are complex numbers; z, τ (Im τ > 0) are complex variables, and H = {τ ∈ ℂ| Im τ > 0}. Further, (\frac{h}{u}) is the generalized Jacobi symbol; \binom{n}{t} is a binomial coefficient; φ(k) is Euler’s function; e(z) = exp 2πiz; η(γ) = 1 if γ ≥ 0 and η(γ) = −1 if γ < 0.

Let

ϑ_{gh}(z|τ; c, N) = ∑_{m≡c (mod N)}(−1)^{h(m−c)/N} ×

× e(\frac{1}{2N} (m + \frac{g}{2})^{2}τ)e((m + \frac{g}{2})z);  (1.1)

hence

\frac{∂^{n}}{∂z^{n}} ϑ_{gh}(z|τ; c, N) = (πi)^{n} ∑_{m≡c (mod N)}(−1)^{h(m−c)/N}(2m + g)^{n} ×

×e(\frac{1}{2N} (m + \frac{g}{2})^{2}τ)e((m + \frac{g}{2})z) (n ≥ 0).  (1.2)

485 Put

1991 Mathematics Subject Classification. 11F37, 11F27, 11F11.

Key words and phrases. Entire modular form, congruence subgroup, multiplier system, theta-function, half integral weight.

1072-947X/96/0900-0485$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

ϑ_{gh}^{(n)}(τ; c, N) = \frac{∂^{n}}{∂z^{n}} ϑ_{gh}(z|τ; c, N)|_{z=0} (n ≥ 0).  (1.3)

It is known (see, for e.g., [1], formulas (11) and (12)) that

ϑ_{g,h+2j}^{(n)}(τ; c, N) = ϑ_{gh}^{(n)}(τ; c, N) (n ≥ 0),  (1.4)

\begin{aligned}ϑ_{gh}^{(n)}(τ + β; c, N) & =e(\frac{β}{2N} (c+ \frac{g}{2})^{2})ϑ_{g,h+βg+βN}^{(n)}(τ; c, N) (n≥0), \\ ϑ_{gh}^{(n)}(τ − β; c, N) & = (−1)^{n}e( − \frac{β}{2N} (c + \frac{g}{2})^{2}) × \\ & × ϑ_{−g,−h+βg−βN}^{(n)}(τ; −c, N) (n ≥ 0). \end{aligned}  (1.5)

From (1.1) and (1.2), according to notation (1.3) it follows, in particular, that

ϑ_{gh}^{(n)}(τ; 0, N) = (πi)^{n} ∑_{m=−∞}^{∞}(−1)^{hm}(2Nm + g)^{n} ×

× e(\frac{1}{2N} (Nm + \frac{g}{2})^{2}τ) (n ≥ 0).  (1.6)

For ξ_{2} ≠ 0, ξ_{1}, g, h, N with ξ_{1}g + ξ_{2}h + ξ_{1}ξ_{2}N ≡ 0 (mod 2), put

S_{gh}( \begin{matrix}ξ_{1} \\ ξ_{2} \end{matrix} ; c, N) = ∑_{\begin{matrix}mmodN|ξ_{2}| \\ m≡c (mod N) \end{matrix}}(−1)^{h(m−c)/N}e(\frac{ξ_{1}}{2Nξ_{2}} (m + \frac{g}{2})^{2}).

Finally, let

Γ = \{\frac{ατ + β}{γτ + δ} | αδ − βγ = 1\},
Γ_{0}(4N) = \{\frac{ατ + β}{γτ + δ} ∈ Γ | γ ≡ 0 (mod 4N)\}.

For τ ∈ H put

(γτ + δ)^{s/2} = ((γτ + δ)^{1/2})^{s}, −\frac{π}{2} < arg(γτ + δ)^{1/2} ≤ \frac{π}{2} .

Definition. Let M be a matrix of an arbitrary substitution from Γ_{0}(4N), and let v(M) be a multiplier system on Γ_{0}(4N) and of weight \frac{s}{2}. We shall say that a function F defined on H is an entire modular form of weight \frac{s}{2} and of multiplier system v(M) on Γ_{0}(4N), if

(1) F is regular on H;

(2) for all matrices M = (\begin{matrix}α & β \\ γ & δ \end{matrix}) of substitutions from Γ_{0}(4N) and all

τ ∈ H,

F (\frac{ατ + β}{γτ + δ} ) = v(M)(γτ + δ)^{s/2}F (τ);

(3) in the neighborhood of the point τ = i∞,

F (τ) = ∑_{m=0}^{∞} A_{m}e(mτ);

(4) for all substitutions from Γ in the neighborhood of each rational point τ = −\frac{δ}{γ} (γ ≠ 0, (γ, δ) = 1),

(γτ + δ)^{s/2}F (τ) = ∑_{m=0}^{∞} A′_{m}e(\frac{m}{4N} \frac{ατ + β}{γτ + δ} ).

Lemma 1 ([1], p. 58, Lemma 3). If g is even, then for n ≥ 0 and all substitutions from Γ_{0}(4N) we have

ϑ_{gh}^{(n)}(\frac{ατ + β}{γτ + δ}; 0, 2N) =

= (sgn δ)^{n}i^{(2n+1)η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2}(\frac{2βN sgn δ}{|δ|} )(γτ + δ)^{(2n+1)/2} ×

×e( − \frac{αγδ^{2}h^{2}}{16N} )e(\frac{βδg^{2}}{4} \frac{δ^{2φ(2N)−2}}{4N} )ϑ_{αg,h}^{(n)}(τ; 0, 2N).

Lemma 2 ([1], p. 61, Lemma 4). If γ ≠ 0, then for n ≥ 0

(γτ + δ)^{(2n+1)/2}ϑ_{gh}^{(n)}(τ; 0, 2N) =

= e((2n + 1) sgn γ/8)(2N|γ|)^{−1/2}( − i sgn γ)^{n} ×

× ∑_{H mod2N} φ_{g′gh}(0, H; 2N)\{ϑ_{g′h′}^{(n)}(\frac{ατ + β}{γτ + δ}; H, 2N) +

+ ∑_{t=1}^{n} \binom{n}{t} ∑_{k=1}^{t} \frac{A_{tk}}{k!} |_{z=0} · ϑ_{g′h′}^{(n−t)}(\frac{ατ + β}{γτ + δ}; H, 2N)\},

where

g′ = δg − γh − 2γδN, h′ = −βg + αh − 2αβN,  (1.7)

\begin{aligned}φ_{g′gh}(0, H; 2N) & = e(\frac{αβ}{4N} (H + \frac{g′}{2} )^{2})e( − \frac{βg}{4N} (H + \frac{g′}{2} )) × \\ & × S_{g−αg′,h−βg′}( \begin{matrix}δ \\ −γ \end{matrix} ; −αH, 2N), \end{aligned}

A_{tk}|_{z=0} = \begin{cases}(2k)!(−2Nγπi(γτ + δ))^{k} & if t = 2k \\ 0 & if t ≠ 2k \end{cases}  (1.8)
(t = 1, 2, …, n; k = 1, 2, …, t).

From (1.8) it follows that

∑_{k=1}^{t} \frac{A_{tk}}{k!} |_{z=0} = \begin{cases}\frac{A_{t,t/2}}{(t/2)!} |_{z=0} & if 2|t, \\ 0 & if 2 ∤ t. \end{cases}  (1.9)

2.

Lemma 3. For given N let

Ω(τ; g_{l}, h_{l}, c_{l}, N_{l}) =

= Ω(τ; g_{1}, g_{2}, g_{3}; h_{1}, h_{2}, h_{3}; c_{1}, c_{2}, c_{3}; N_{1}, N_{2}, N_{3}) =

= \{\frac{1}{N_{1}} ϑ′′_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})ϑ_{g_{2}h_{2}}(τ; c_{2}, 2N_{2}) +

−\frac{1}{N_{2}} ϑ′′_{g_{2}h_{2}}(τ; c_{2}, 2N_{2})ϑ_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})\}ϑ_{g_{3}h_{3}}(τ; c_{3}, 2N_{3}), (2.1)

where

2|g_{k}, N_{k}|N (k = 1, 2, 3), 4|N ∑_{k=1}^{3} \frac{h_{k}}{N_{k}} .  (2.2)

Then for all substitutions from Γ in the neighborhood of each rational point τ = −\frac{δ}{γ} (γ ≠ 0, (γ, δ) = 1), we have

(γτ + δ)^{7/2}Ω(τ; g_{l}, h_{l}, 0, N_{l}) = ∑_{n=0}^{∞} G_{n}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ).  (2.3)

Proof. 1. From Lemma 2 for n = 2 and n = 0 (respectively with g_{1}, h_{1}, N_{1}, g′_{1}, h′_{1}, H_{1} and g_{2}, h_{2}, N_{2}, g′_{2}, h′_{2}, H_{2} instead of g, h, N, g′, h′, H), according to (1.8)–(1.9), it follows that

\frac{1}{N_{1}} (γτ + δ)^{3}ϑ′′_{g_{1}h_{1}}(τ; 0, 2N_{1})ϑ_{g_{2}h_{2}}(τ; 0, 2N_{2}) =

= −e(\frac{3}{4} sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2} ×

× ∑_{\begin{matrix}H_{1} mod2N_{1} \\ H_{2} mod2N_{2} \end{matrix}} φ_{g′_{1}g_{1}h_{1}}(0, H_{1}; 2N_{1})φ_{g′_{2}g_{2}h_{2}}(0, H_{2}; 2N_{2}) ×

×\{\frac{1}{N_{1}} ϑ′′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})ϑ_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2}) +

−4γπi(γτ + δ)ϑ_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1}) ×

×ϑ_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2})\}.  (2.4)

Replacing N_{1}, g_{1}, h_{1}, H_{1}, g′_{1}, h′_{1} by N_{2}, g_{2}, h_{2}, H_{2}, g′_{2}, h′_{2} in (2.4), and vice versa, we obtain

\frac{1}{N_{2}} (γτ + δ)^{3}ϑ′′_{g_{2}h_{2}}(τ; 0, 2N_{2})ϑ_{g_{1}h_{1}}(τ; 0, 2N_{1}) =

= −e(\frac{3}{4} sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2} ×

× ∑_{\begin{matrix}H_{1} mod2N_{1} \\ H_{2} mod2N_{2} \end{matrix}} φ_{g′_{2}g_{2}h_{2}}(0, H_{2}; 2N_{2})φ_{g′_{1}g_{1}h_{1}}(0, H_{1}; 2N_{1}) ×

×\{\frac{1}{N_{2}} ϑ′′_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2})ϑ_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1}) +

−4γπi(γτ + δ)ϑ_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2}) ×

×ϑ_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})\}.  (2.5)

Subtracting (2.5) from (2.4), we obtain

(γτ + δ)^{3}\{\frac{1}{N_{1}} ϑ′′_{g_{1}h_{1}}(τ; 0, 2N_{1})ϑ_{g_{2}h_{2}}(τ; 0, 2N_{2}) +

−\frac{1}{N_{2}} ϑ′′_{g_{2}h_{2}}(τ; 0, 2N_{2})ϑ_{g_{1}h_{1}}(τ; 0, 2N_{1})\} =

= −e(\frac{3}{4} sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2} ×

× ∑_{\begin{matrix}H_{1} mod2N_{1} \\ H_{2} mod2N_{2} \end{matrix}} φ_{g′_{1}g_{1}h_{1}}(0, H_{1}; 2N_{1})φ_{g′_{2}g_{2}h_{2}}(0, H_{2}; 2N_{2}) ×

×\{\frac{1}{N_{1}} ϑ′′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})ϑ_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2}) +

−\frac{1}{N_{2}} ϑ′′_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2})ϑ_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})\}. (2.6)

From Lemma 2 for n = 0, we have

(γτ + δ)^{1/2}ϑ_{g_{3}h_{3}}(τ; 0, 2N_{3}) = e( sgn γ/8)(2N_{3}|γ|)^{−1/2} ×

× ∑_{H_{3} mod2N_{3}} φ_{g′_{3}g_{3}h_{3}}(0, H_{3}; 2N_{3})ϑ_{g′_{3}h′_{3}}(\frac{ατ + β}{γτ + δ}; H_{3}, 2N_{3}).  (2.7)

Multiplying (2.6) by (2.7), according to (2.1), we obtain

(γτ + δ)^{7/2}Ω(τ; g_{l}, h_{l}, 0, N_{l}) =
= −e(7 sgn γ/8)(8N_{1}N_{2}N_{3}|γ|^{3})^{−1/2} ×
× ∑_{H_{1} mod2N_{1} ; H_{2} mod2N_{2} ; H_{3} mod2N_{3}} ∏_{k=1}^{3} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})Ω(\frac{ατ + β}{γτ + δ}; g′_{l}, h′_{l}, H_{l}, N_{l}). (2.8)

2. In (2.8) let γ be even. Then, by (1.7), g′_{k} (k = 1, 2, 3) are also even. Now in (1.2) and (1.3) instead of m let us introduce new letters of summations m_{k} defined by the equalities m − H_{k} = 2N_{k}m_{k} (k = r, t, 3), respectively, and for brevity we put

T_{k} = (H_{k} + 2N_{k}m_{k} + \frac{g′_{k}}{2} )^{2} (k = r, t, 3).

Then, by (2.1), in (2.8) for r = 1, t = 2 and r = 2, t = 1, we get

ϑ′′_{g′_{r}h′_{r}}(\frac{ατ + β}{γτ + δ}; H_{r}, 2N_{r}) ∏_{k=t,3} ϑ_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ}; H_{k}, 2N_{k}) =

= (πi)^{2} ∑_{m_{r}=−∞}^{∞}(−1)^{h′_{r}m_{r}}(2(H_{r} + 2N_{r}m_{r}) + g′_{r})^{2}e(\frac{N/N_{r}}{4N} T_{r} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{m_{t}=−∞}^{∞}(−1)^{h′_{t}m_{t}}e(\frac{N/N_{t}}{4N} T_{t} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{m_{3}=−∞}^{∞}(−1)^{h′_{3}m_{3}}e(\frac{N/N_{3}}{4N} T_{3} \frac{ατ + β}{γτ + δ} )=

= ∑_{n_{r}=0}^{∞} B_{n_{r}}e(\frac{n_{r}}{4N} \frac{ατ + β}{γτ + δ} ) ∑_{n_{t}=0}^{∞} B_{n_{t}}e(\frac{n_{t}}{4N} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{n_{3}=0}^{∞} B_{n_{3}}e(\frac{n_{3}}{4N} \frac{ατ + β}{γτ + δ} ) = ∑_{n=0}^{∞} C_{n}^{(r,t)}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ),  (2.9)

since by (2.2) n_{k} = \frac{N}{N_{k}} T_{k} are non-negative integers. Thus, for even γ, (2.3) follows from (2.1),(2.8), and (2.9).

In (2.8) let now γ be odd. If h_{1}, h_{2}, h_{3} are even, then by (1.7), g′_{1}, g′_{2}, g′_{3} are also even, and we obtain the same result. But if h_{r} is odd, then by (1.7), h′_{r} and g′_{r} will also be odd, and in (1.2) we shall have

(m + g′_{r}/2)^{2} = (m + (g′_{r} − 1)/2)^{2} + (m + (g′_{r} − 1)/2) + 1/4.

Hence

ϑ′′_{g′_{r}h′_{r}}(\frac{ατ + β}{γτ + δ}; H_{r}, 2N_{r}) = (πi)^{2}e(\frac{h′_{r}}{16N_{r}} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{m_{r}=−∞}^{∞}(−1)^{h′_{r}m_{r}}(2(H_{r} + 2N_{r}m_{r})+g′_{r})^{2}e(\frac{N/N_{r}}{4N} W_{r} \frac{ατ + β}{γτ + δ} ) (r =1, 2).

Here by (1.4) we can imply that h′_{r} = 1, since by (1.7), h′_{r} and h_{r} have the same parity, and

W_{r} = (H_{r} + 2N_{r}m_{r} + (g′_{r} − 1)/2)^{2} +
+ (H_{r} + 2N_{r}m_{r} + (g′_{r} − 1)/2) (r = 1, 2).  (2.10)

Analogously,

ϑ_{g′_{t}h′_{t}}(\frac{ατ + β}{γτ + δ}; H_{t}, 2N_{t}) = e(\frac{h′_{t}}{16N_{t}} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{m_{t}=−∞}^{∞}(−1)^{h′_{t}m_{t}}e(\frac{N/N_{t}}{4N} W_{t} \frac{ατ + β}{γτ + δ} ) (t = 2, 1).

The same formula is valid for t = 3 with the same remarks concerning h′_{3} as for h′_{t}, t = 2, 1. Further, for k = t, 3 we have

W_{k} = \begin{cases}T_{k} & if 2|g′_{k}, \\ (H_{k} + 2N_{k}m_{k} + (g′_{k} − 1)/2)^{2}+ \\ (H_{k} + 2N_{k}m_{k} + (g′_{k} − 1)/2) & if 2 ∤ g′_{k}. \end{cases}  (2.11)

Thus, if among h′_{1}, h′_{2}, h′_{3} at least one is odd, then, as in (2.9), for r = 1, t = 2 and r = 2, t = 1 we have

ϑ′′_{g′_{r}h′_{r}}(\frac{ατ + β}{γτ + δ}; H_{r}, 2N_{r}) ∏_{k=t,3} ϑ_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ}; H_{k}, 2N_{k}) =

= e(\frac{1}{4N} (\frac{1}{4} N ∑_{k=1}^{3} \frac{h′_{k}}{N_{k}} )\frac{ατ + β}{γτ + δ} ) ∑_{n_{r}=0}^{∞} B′_{n_{r}}e(\frac{n_{r}}{4N} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{n_{t}=0}^{∞} B′_{n_{t}}e(\frac{n_{t}}{4N} \frac{ατ + β}{γτ + δ} ) ∑_{n_{3}=0}^{∞} B′_{n_{3}}e(\frac{n_{3}}{4N} \frac{ατ + β}{γτ + δ} ) =

= e(\frac{1}{4N} (\frac{1}{4} N ∑_{k=1}^{3} \frac{h′_{k}}{N_{k}} )\frac{ατ + β}{γτ + δ} ) ∑_{n′=0}^{∞} D_{n′}^{(r,t)}e(\frac{n′}{4N} \frac{ατ + β}{γτ + δ} ) =

= ∑_{n=0}^{∞} D_{n}^{(r,t)}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ),  (2.12)

since, by (2.2), (2.10), and (2.11), n_{k} = \frac{N}{N_{k}} W_{k} and n = \frac{1}{4}N ∑_{k=1}^{3} \frac{h′_{k}}{N_{k}} + n′ are

non-negative integers. Thus, for odd γ, (2.3) follows from (2.1), (2.8), and (2.12).

Theorem 1. For given N the function Ω(τ; g_{l}, h_{l}, 0, N_{l}) is an entire modular form of weight 7/2 and of the multiplier system

v(M) = i^{3η(γ)(sgn δ−1)/2}i^{3(|δ|−1)^{2}/4}(\frac{βΔ sgn δ}{|δ|} )  (2.13)

(M = (\begin{matrix}α & β \\ γ & δ \end{matrix}) is a matrix of the substitution from Γ_{0}(4N) and Δ is the

determinant of an arbitrary positive quadratic form with integer coefficients in 7 variables) on Γ_{0}(4N) if the following conditions hold:

(1) 2|g_{k}, N_{k}|N (k = 1, 2, 3), (2.14)

(2) 4|N ∑_{k=1}^{3} \frac{h_{k}^{2}}{N_{k}} 4| ∑_{k=1}^{s} \frac{g_{k}^{2}}{4N_{k}}, (2.15)

3) for all α and δ with αδ ≡ 1 (mod 4N)

(\frac{N_{1}N_{2}N_{3}}{|δ|} )Ω(τ; αg_{l}, h_{l}, 0, N_{l}) = (\frac{Δ}{|δ|})Ω(τ; g_{l}, h_{l}, 0, N_{l}). (2.16)

Proof. 1. It is well known that theta-series (1.1)–(1.2) are regular on H; hence the function Ω(τ; g_{l}, h_{l}, 0, N_{l}) satisfies condition (1) of the Definition.

2. It is easily verified that (2.15) implies

4|Nδ^{2} ∑_{k=1}^{3} h_{k}^{2}/N_{k}, 4| ∑_{k=1}^{3} δ^{2φ(2N_{k})−2}g_{k}^{2}/4N_{k},  (2.17)

since 2 ∤ δ, because αδ ≡ 1 (mod 4N).

From (2.14) it follows that

Γ_{0}(4N) ⊂ Γ_{0}(4N_{k}) (k = 1, 2, 3).  (2.18)

By Lemma 1 for n = 2 and n = 0, according to (2.17) and (2.18), for all substitutions from Γ_{0}(4N), we have

ϑ′′_{g_{r}h_{r}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{r}) = i^{5η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2} ×

×(\frac{2βN_{r} sgn δ}{|δ|} )(γτ + δ)^{5/2}ϑ′′_{αg_{r},h_{r}}(τ; 0, 2N_{r}),  (2.19)

ON SOME ENTIRE MODULAR FORMS OF WEIGHTS 493

ϑ_{g_{t}h_{t}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{t}) = i^{η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2} ×

×(\frac{2βN_{t} sgn δ}{|δ|} )(γτ + δ)^{1/2}ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t}),  (2.20)

ϑ_{g_{3}h_{3}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{3}) = i^{η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2} ×

×(\frac{2βN_{3} sgn δ}{|δ|} )(γτ + δ)^{1/2}ϑ_{αg_{3},h_{3}}(τ; 0, 2N_{3}).  (2.21)

Thus for all substitutions from Γ_{0}(4N), if r = 1, t = 2 and r = 2, t = 1, we have

ϑ′′_{g_{r}h_{r}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{r}) ∏_{k=t,3} ϑ_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{k}) =

= i^{7η(γ)(sgn δ−1)/2}i^{3(1−|δ|)/2}(\frac{2β sgn δ}{|δ|} )(\frac{N_{r}N_{t}N_{3}}{|δ|} )(γτ + δ)^{7/2} ×

×ϑ′′_{αg_{r},h_{r}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{r}) ∏_{k=t,3} ϑ_{αg_{k},h_{k}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{k}). (2.22)

It is not difficult to verify that

i^{3(1−|δ|)/2}(\frac{2}{|δ|}) = i^{3(|δ|−1)^{2}/4}.  (2.23)

Hence, by (2.1), (2.22), (2.23), (2.13), and (2.16) we get

Ω(\frac{ατ + β}{γτ + δ}; g_{l}, h_{l}, 0, N_{l}) = i^{3η(γ)(sgn δ−1)/2}i^{3(|δ|−1)^{2}/4} ×

×(\frac{β sgn δ}{|δ|} )(\frac{N_{1}N_{2}N_{3}}{|δ|} )(γτ + δ)^{7/2}Ω(τ; αg_{l}, h_{l}, 0, N_{l}) =

= v(M)(γτ + δ)^{7/2}Ω(τ; g_{l}, h_{l}, 0, N_{l}).

Thus, the function Ω(τ; g_{l}, h_{l}, 0, N_{l}) satisfies condition (2) of the Definition.

3. From (1.6) it follows for r = 1, t = 2 and r = 2, t = 1 that

ϑ′′_{g_{r}h_{r}}(τ; 0, 2N_{r})ϑ_{g_{t}h_{t}}(τ; 0, 2N_{t})ϑ_{g_{3}h_{3}}(τ; 0, 2N_{3}) =

= (πi)^{2} ∑_{m_{r},m_{t},m_{3}=−∞}^{∞}(−1)^{h_{r}m_{r}+h_{t}m_{t}+h_{3}m_{3}}(4N_{r}m_{r} + g_{r})^{2} ×

×e( ∑_{k=r,t,3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} B_{n}^{(r,t)}e(nτ),  (2.24)

where

n = ∑_{k=1}^{3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2} =

= ∑_{k=1}^{3}(N_{k}m_{k}^{2} + m_{k}g_{k}/2) + \frac{1}{4} ∑_{k=1}^{3} g_{k}^{2}/4N_{k},

by (2.14) and (2.15), is a non-negative integer. Thus, by (2.1) and (2.24), the function Ω(τ; g_{l}, h_{l}, 0, N_{l}) satisfies condition (3) of the Definition.

4. By Lemma 3, the function Ω(τ; g_{l}, h_{l}, 0, N_{l}) satisfies condition (4) of the Definition.

3.

Lemma 4. For given N let

Ψ_{1}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = Ψ_{1}(τ; g_{1}, g_{2}, g_{3}; h_{1}, h_{2}, h_{3}; c_{1}, c_{2}, c_{3}; N_{1}, N_{2}, N_{3}) =

= \{\frac{1}{N_{1}} ϑ′′_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})ϑ_{g_{2}h_{2}}(τ; c_{2}, 2N_{2}) +

−\frac{1}{N_{2}} ϑ′′_{g_{2}h_{2}}(τ; c_{2}, 2N_{2})ϑ_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})\}ϑ′_{g_{3}h_{3}}(τ; c_{3}, 2N_{3})  (3.1)

and

Ψ_{2}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = Ψ_{2}(τ; g_{1}, g_{2}, g_{3}; h_{1}, h_{2}, h_{3}; c_{1}, c_{2}, c_{3}; N_{1}, N_{2}, N_{3}) =

= \{\frac{1}{N_{1}} ϑ′′′_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})ϑ_{g_{2}h_{2}}(τ; c_{2}, 2N_{2}) +

−\frac{3}{N_{2}} ϑ′′_{g_{2}h_{2}}(τ; c_{2}, 2N_{2})ϑ′_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})\}ϑ_{g_{3}h_{3}}(τ; c_{3}, 2N_{3}), (3.2)

where

2|g_{k}, N_{k}|N (k = 1, 2, 3), 4|N ∑_{k=1}^{3} \frac{h_{k}}{N_{k}} .  (3.3)

Then for all substitutions from Γ, in the neighborhood of each rational point τ = −\frac{δ}{γ} (γ ≠ 0, (γ, δ) = 1), we have

(γτ + δ)^{9/2}Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l})= ∑_{n=0}^{∞}G_{n}^{(j)}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ) (j =1, 2). (3.4)

Proof. 1. Formula (2.6) is obtained in Lemma 3. From Lemma 2, for n = 1, we have

(γτ + δ)^{3/2}ϑ′_{g_{3}h_{3}}(τ; 0, 2N_{3}) = −e(3 sgn γ/8)(2N_{3}|γ|)^{−1/2}(i sgn γ) ×

× ∑_{H_{3} mod2N_{3}} φ_{g′_{3}g_{3}h_{3}}(0, H_{3}; 2N_{3})ϑ′_{g′_{3}h′_{3}}(\frac{ατ + β}{γτ + δ}; H_{3}, 2N_{3}).  (3.5)

Multiplying (2.6) by (3.5), according to (3.1), we obtain

(γτ + δ)^{9/2}Ψ_{1}(τ; g_{l}, h_{l}, 0, N_{l}) =
= e(9 sgn γ/8)(8N_{1}N_{2}N_{3}|γ|^{3})^{−1/2}(i sgn γ) ×
× ∑_{H_{1} mod2N_{1} ; H_{2} mod2N_{2} ; H_{3} mod2N_{3}} ∏_{k=1}^{3} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})Ψ_{1}(\frac{ατ + β}{γτ + δ}; g′_{l}, h′_{l}, H_{l}, N_{l}). (3.6)

Reasoning further just as in Lemma 3.2, but taking everywhere ϑ′_{g_{3}h_{3}}(\frac{ατ+β}{γτ+δ}; H_{3}, 2N_{3}) instead of ϑ_{g_{3}h_{3}}(\frac{ατ+β}{γτ+δ}; H_{3}, 2N_{3}), from (3.1) and (3.6) we obtain (3.4) if j = 1.

2. From Lemma 2 for n = 3, n = 0, and respectively for n = 2, n = 1, it follows that

\frac{1}{N_{1}} (γτ + δ)^{4}ϑ′′′_{g_{1}h_{1}}(τ; 0, 2N_{1})ϑ_{g_{2}h_{2}}(τ; 0, 2N_{2}) =

= \frac{1}{N_{1}} e(sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2}(i sgn γ) ×

× ∑_{H_{1} mod2N_{1}} φ_{g′_{1}g_{1}h_{1}}(0, H_{1}; 2N_{1})\{ϑ′′′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1}) +

+ ∑_{t=1}^{3} \binom{3}{t} ∑_{k=1}^{t} \frac{A_{tk}}{k!} |_{z=0}ϑ_{g′_{1}h′_{1}}^{(3−t)}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})\} ×

× ∑_{H_{2} mod2N_{2}} φ_{g′_{2}g_{2}h_{2}}(0, H_{2}; 2N_{2})ϑ_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2}) =

= e(sgn γ)(i sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2} ∑_{\begin{matrix}H_{1} mod2N_{1} \\ H_{2} mod2N_{2} \end{matrix}} φ_{g′_{1}g_{1}h_{1}}(0, H_{1}; 2N_{1}) ×

×φ_{g′_{2}g_{2}h_{2}}(0, H_{2}; 2N_{2})\{\frac{1}{N_{1}} ϑ′′′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})ϑ_{g′_{2}h′_{2}}(τ; 0, 2N_{2}) +

−12γπi(γτ + δ)ϑ′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})ϑ_{g′_{2}h′_{2}}(τ; 0, 2N_{2})\} (3.7)

and

\frac{3}{N_{2}} (γτ + δ)^{4}ϑ′′_{g_{2}h_{2}}(τ; 0, 2N_{2})ϑ′_{g_{1}h_{1}}(τ; 0, 2N_{1}) =

= e(sgn γ)(i sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2} ∑_{\begin{matrix}H_{1} mod2N_{1} \\ H_{2} mod2N_{2} \end{matrix}} φ_{g′_{1}g_{1}h_{1}}(0, H_{1}; 2N_{1}) ×

×φ_{g′_{2}g_{2}h_{2}}(0, H_{2}; 2N_{2})\{\frac{3}{N_{2}} ϑ′′_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2}) +

−12γπi(γτ + δ)ϑ′_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2})\}ϑ′_{g′_{1}h′_{1}}(τ; 0, 2N_{1}). (3.8)

Subtracting (3.8) from (3.7) and multiplying the obtained result by (2.7), according to (3.2), we get

(γτ + δ)^{9/2}Ψ_{2}(τ; g_{l}, h_{l}, 0, N_{l}) =
= e(9 sgn γ/8)(8N_{1}N_{2}N_{3}|γ|^{3})^{−1/2}(i sgn γ) ×
× ∑_{H_{1} mod2N_{1} ; H_{2} mod2N_{2} ; H_{3} mod2N_{3}} ∏_{k=1}^{3} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})Ψ_{2}(\frac{ατ + β}{γτ + δ}; g′_{l}, h′_{l}, H_{l}, N_{l}). (3.9)

In (3.9) let γ be even. Then, reasoning as in Lemma 3.2, by (3.2), from (3.9) we obtain

\{\frac{1}{N_{1}} ϑ′′′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})ϑ_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2}) +

−\frac{3}{N_{2}} ϑ′′_{g′_{2}h′_{2}}(\frac{ατ + β}{γτ + δ}; H_{2}, 2N_{2})ϑ′_{g′_{1}h′_{1}}(\frac{ατ + β}{γτ + δ}; H_{1}, 2N_{1})\} ×

×ϑ_{g′_{3}h′_{3}}(\frac{ατ + β}{γτ + δ}; H_{3}, 2N_{3}) = \{\frac{(πi)^{3}}{N_{1}} ×

× ∑_{m_{1}=−∞}^{∞}(−1)^{h′_{1}m_{1}}(2(H_{1} + 2N_{1}m_{1}) + g′_{1})^{3}e(\frac{N/N_{1}}{4N} T_{1} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{m_{2}=−∞}^{∞}(−1)^{h′_{2}m_{2}}e(\frac{N/N_{2}}{4N} T_{2} \frac{ατ + β}{γτ + δ} ) +

−\frac{3(πi)^{3}}{N_{1}} ∑_{m_{2}=−∞}^{∞}(−1)^{h′_{2}m_{2}}(2(H_{2} + 2N_{2}m_{2}) + g′_{2})^{2}e(\frac{N/N_{2}}{4N} T_{2} \frac{ατ + β}{γτ + δ} ) ×

× ∑_{m_{1}=−∞}^{∞}(−1)^{h′_{1}m_{1}}(2(H_{1} + 2N_{1}m_{1}) + g′_{1})e(\frac{N/N_{1}}{4N} T_{1} \frac{ατ + β}{γτ + δ} )\} ×

× ∑_{m_{3}=−∞}^{∞}(−1)^{h′_{3}m_{3}}e(\frac{N/N_{3}}{4N} T_{3} \frac{ατ + β}{γτ + δ} ) =

= \{ ∑_{n_{1}=0}^{∞} B_{n_{1}}e(\frac{n_{1}}{4N} \frac{ατ + β}{γτ + δ} ) ∑_{n_{2}=0}^{∞} B_{n_{2}}e(\frac{n_{2}}{4N} \frac{ατ + β}{γτ + δ} ) +

− ∑_{n_{2}=0}^{∞} B′_{n_{2}}e(\frac{n_{2}}{4N} \frac{ατ + β}{γτ + δ} ) ∑_{n_{1}=0}^{∞} B′_{n_{1}}e(\frac{n_{1}}{4N} \frac{ατ + β}{γτ + δ} )\} ×

× ∑_{n_{3}=0}^{∞} B_{n_{3}}e(\frac{n_{3}}{4N} \frac{ατ + β}{γτ + δ} ) = ∑_{n=0}^{∞} C_{n}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ).  (3.10)

Here T_{k} and n_{k} have the same meaning as in Lemma 3.2.

Thus, for even γ, (3.4) follows from (3.2), (3.9) and (3.10) if j = 2.

Further, reasoning as in Lemma 3.2, we obtain (3.4) for odd γ if j = 2.

Theorem 2. For given N the functions Ψ_{1}(τ; g_{l}, h_{l}, 0, N_{l}) and Ψ_{2}(τ; g_{l}, h_{l}, 0, N_{l}) are entire modular forms of weight 9/2 and of the multiplier system

v(M) = i^{η(γ)(sgn δ−1)/2}i^{(|δ|−1)^{2}/4}(\frac{βΔ sgn δ}{|δ|} )  (3.11)

(M = (\begin{matrix}α & β \\ γ & δ \end{matrix}) is the matrix of the substitution from Γ_{0}(4N) and Δ is the

determinant of an arbitrary positive quadratic form with integer coefficients in 9 variables) on Γ_{0}(4N) if the following conditions hold:

(1) 2|g_{k}, N_{k}|N (k = 1, 2, 3), (3.12)

(2) 4|N ∑_{k=1}^{3} \frac{h_{k}^{2}}{N_{k}} 4| ∑_{k=1}^{s} \frac{g_{k}^{2}}{4N_{k}}, (3.13)

(3) for all α and δ with αδ ≡ 1 (mod 4N)

sgn δ(\frac{N_{1}N_{2}N_{3}}{|δ|} )Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) =

= (\frac{−Δ}{|δ|})Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2).  (3.14)

Proof. 1. As in the case of Theorem 1, the functions Ψ_{1}(τ; g_{l}, h_{l}, 0, N_{l}) and Ψ_{2}(τ; g_{l}, h_{l}, 0, N_{l}) satisfy condition (1) and, by Lemma 4, also condition (4) of the Definition.

2. By Lemma 1 for n = 3 and n = 1, according to (2.17) and (2.18), for all substitutions from Γ_{0}(4N), we have

ϑ′′′_{g_{1}h_{1}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{1}) = sgn δi^{7η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2} ×

498 G. LOMADZE

×(\frac{2βN_{1} sgn δ}{|δ|} )(γτ + δ)^{7/2}ϑ′′′_{αg_{1},h_{1}}(τ; 0, 2N_{1}),  (3.15)

ϑ′_{g_{1}h_{1}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{1}) = sgn δi^{3η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2} ×

×(\frac{2βN_{1} sgn δ}{|δ|} )(γτ + δ)^{3/2}ϑ′_{αg_{1},h_{1}}(τ; 0, 2N_{1}),  (3.16)

ϑ′_{g_{3}h_{3}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{3}) = sgn δi^{3η(γ)(sgn δ−1)/2}i^{(1−|δ|)/2} ×

×(\frac{2βN_{3} sgn δ}{|δ|} )(γτ + δ)^{3/2}ϑ′_{αg_{3},h_{3}}(τ; 0, 2N_{3}).  (3.17)

Thus, for all substitutions from Γ_{0}(4N) we respectively get:

(1) by (2.19), (2.20), and (3.17)

ϑ′′_{g_{r}h_{r}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{r})ϑ_{g_{t}h_{t}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{t})ϑ′_{g_{3}h_{3}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{3}) =

= sgn δi^{η(γ)(sgn δ−1)/2}i^{3(1−|δ|)/2}(\frac{2β sgn δ}{|δ|} )(\frac{N_{1}N_{2}N_{3}}{|δ|} ) ×

×(γτ + δ)^{9/2}ϑ′′_{αg_{r},h_{r}}(τ; 0, 2N_{r})ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t})ϑ′_{αg_{3},h_{3}}(τ; 0, 2N_{3}) (3.18)

for r = 1, t = 2 and r = 2, t = 1; (2) by (3.15), (2.20) for t = 2 and (2.21)

ϑ′′′_{g_{1}h_{1}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{1})ϑ_{g_{2}h_{2}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{2})ϑ_{g_{3}h_{3}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{3}) =

= sgn δi^{η(γ)(sgn δ−1)/2}i^{3(1−|δ|)/2}(\frac{2β sgn δ}{|δ|} )(\frac{N_{1}N_{2}N_{3}}{|δ|} ) ×

×(γτ + δ)^{9/2}ϑ′′′_{αg_{1},h_{1}}(τ; 0, 2N_{1})ϑ_{αg_{2},h_{2}}(τ; 0, 2N_{2})ϑ_{αg_{3},h_{3}}(τ; 0, 2N_{3}); (3.19)

(3) by (2.19) for r = 2 , (3.16), and (2.21)

ϑ′′_{g_{2}h_{2}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{2})ϑ′_{g_{1}h_{1}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{1})ϑ_{g_{3}h_{3}}(\frac{ατ + β}{γτ + δ}; 0, 2N_{3}) =

= sgn δi^{η(γ)(sgn δ−1)/2}i^{3(1−|δ|)/2}(\frac{2β sgn δ}{|δ|} )(\frac{N_{1}N_{2}N_{3}}{|δ|} ) ×

×(γτ + δ)^{9/2}ϑ′′_{αg_{2},h_{2}}(τ; 0, 2N_{2})ϑ′_{αg_{1},h_{1}}(τ; 0, 2N_{1})ϑ_{αg_{3},h_{3}}(τ; 0, 2N_{3}). (3.20)

It is not difficult to verify that

i^{3(1−|δ|)/2}(\frac{−2}{|δ|}) = i^{(|δ|−1)^{2}/4}.  (3.21)

Hence, by (3.1), (3.18), (3.21), (3.11), (3.14) (if j = 1) and (3.2), (3.19)– (3.21), (3.11), (3.14) (if j = 2) we get

Ψ_{j}(\frac{ατ + β}{γτ + δ}; g_{l}, h_{l}, 0, N_{l}) = sgn δi^{η(γ)(sgn δ−1)/2}i^{(|δ|−1)^{2}/4} ×

×(\frac{−β sgn δ}{|δ|} )(\frac{N_{1}N_{2}N_{3}}{|δ|} )(γτ + δ)^{9/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) =

= v(M)(γτ + δ)^{9/2}Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2).

Thus the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2) satisfy condition (2) of the Definition.

From (1.6) it follows that

ϑ′′_{g_{r}h_{r}}(τ; 0, 2N_{r})ϑ_{g_{t}h_{t}}(τ; 0, 2N_{t})ϑ′_{g_{3}h_{3}}(τ; 0, 2N_{3}) =

= (πi)^{3} ∑_{m_{r},m_{t},m_{3}=−∞}^{∞}(−1)^{∑_{k=r,t,3} h_{k}m_{k}}(4N_{r}m_{r} + g_{r})^{2}(4N_{3}m_{3} + g_{3}) ×

×e( ∑_{k=r,t,3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} C_{n}^{(r,t)}e(nτ)

for r = 1, t = 2 and r = 2, t = 1,

ϑ′′′_{g_{1}h_{1}}(τ; 0, 2N_{1})ϑ_{g_{2}h_{2}}(τ; 0, 2N_{2})ϑ_{g_{3}h_{3}}(τ; 0, 2N_{3}) =

= (πi)^{3} ∑_{m_{1},m_{2},m_{3}=−∞}^{∞}(−1)^{∑_{k=1}^{3} h_{k}m_{k}}(4N_{1}m_{1} + g_{1})^{3} ×

×e( ∑_{k=1}^{3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} C_{n}e(nτ)

and

ϑ′′_{g_{2}h_{2}}(τ; 0, 2N_{2})ϑ′_{g_{1}h_{1}}(τ; 0, 2N_{1})ϑ_{g_{3}h_{3}}(τ; 0, 2N_{3}) =

= (πi)^{3} ∑_{m_{1},m_{2},m_{3}=−∞}^{∞}(−1)^{∑_{k=1}^{3} h_{k}m_{k}}(4N_{2}m_{2} + g_{2})^{2}(4N_{1}m_{1} + g_{1}) ×

×e( ∑_{k=1}^{3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} D_{n}e(nτ),

since in all these expansions

n = ∑_{k=1}^{3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2},

as has been shown by Theorem 1.3, is a non-negative integer. Therefore, in view of (3.1) and (3.2), the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2) satisfy condition (3) of the Definition.

References

1. G. Lomadze, On some entire modular forms of weights 5 and 6 for the congruence group Γ_{0}(4N). Georgian Math. J. 1(1994), No. 1, 53–76.

(Received 24.11.1994)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia