GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 5, 1999, 471-488

ON SOME ENTIRE MODULAR FORMS OF AN INTEGRAL WEIGHT FOR THE CONGRUENCE

SUBGROUP Γ_{0}(4N)

G. LOMADZE

Abstract. Four types of entire modular forms of weight \frac{s}{2} are constructed for the congruence subgroup Γ_{0}(4N) when s is even. One can find these forms helpful in revealing the arithmetical meaning of additional terms in the formulas for the number of representations of positive integers by positive quadratic forms with integral coefficients in an even number of variables.

All the notations, definitions and lemmas used in this paper are taken from [1].

To make the notations shorter throughout the paper we put

Ψ_{j}(τ; g_{l}, h_{l}, c_{l}, N_{l}) =

= Ψ_{j}(τ; g_{1}, h_{1}, c_{1}, N_{1}; g_{2}, h_{2}, c_{2}, N_{2}; …; g_{s−8}, h_{s−8}, c_{s−8}, N_{s−8}) (j = 1, 2, 3, 4).

1.

Lemma 1. For even s ≥ 12 and given N let

Ψ_{1}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = \{\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})\} ∏_{k=3}^{s−8} ϑ_{g_{k}h_{k}}(τ; c_{k}, 2N_{k}) (1.1)

1991 Mathematics Subject Classification. 11F11, 11F27. Key words and phrases. Positive quadratic form, entire modular form, theta functions

with characteristics, congruence subgroup.

471

1072-947X/99/0900-0471$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

and

Ψ_{2}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = \{\frac{1}{N_{1}^{2}} ϑ_{g_{1}h_{1}}^{(4)}(τ; c_{1}, 2N_{1})ϑ_{g_{2}h_{2}}(τ; c_{2}, 2N_{2}) +

+\frac{1}{N_{2}^{2}} ϑ_{g_{2}h_{2}}^{(4)}(τ; c_{2}, 2N_{2})ϑ_{g_{1}h_{1}}(τ; c_{1}, 2N_{1}) −

−\frac{6}{N_{1}N_{2}} ϑ′′_{g_{1}h_{1}}(τ; c_{1}, 2N_{1})ϑ′′_{g_{2}h_{2}}(τ; c_{2}, 2N_{2})\} ∏_{k=3}^{s−8} ϑ_{g_{k}h_{k}}(τ; c_{k}, 2N_{k}), (1.2)

where

2|g_{k}, N_{k}|N (k = 1, 2, …, s − 8), 4|N ∑_{k=1}^{s−8} \frac{h_{k}}{N_{k}} .  (1.3)

Then we have

(γτ + δ)^{s/2}Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) = ∑_{n=0}^{∞} B_{n}^{(j)}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ) (j = 1, 2) (1.4)

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

In [2] this lemma is proved for arbitrary s > 9 (even and odd).

Lemma 2. For even s we have

i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2} = \begin{cases}1 & if s ≡ 0 (mod 4), \\ sgn δ(\frac{−1}{|δ|}) & if s ≡ 2 (mod 4). \end{cases}

Proof. (a) Let s ≡ 0 (mod 4). Then

i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2} = i^{2hη(γ)(sgn δ−1)} · i^{(2h−4)(1−|δ|)} =

= (−1)^{hη(γ)(sgn δ−1)} · (−1)^{(h−2)(1−|δ|)} = 1.

(b) Let s ≡ 2 (mod 4). Then

i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2} = i^{(4h+2)η(γ)(sgn δ−1)/2} · i^{(4h−6)(1−|δ|)/2} =

= i^{(2h+1)η(γ)(sgn δ−1)} · (−1)^{(2h−3)(1−|δ|)/2} = sgn δ(\frac{−1}{|δ|}).

Theorem 1. For even s and given N the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2) are entire modular forms of weight \frac{s}{2} and the character

χ(δ) = \begin{cases}(\frac{Δ}{|δ|}) & if s ≡ 0 (mod 4), \\ sgn δ(\frac{−Δ}{|δ|}) & if s ≡ 2 (mod 4) \end{cases}  (1.5)

(Δ is the determinant of the positive quadratic form with integral coefficients in s variables for which the function Ψ_{j} will be used) for the congruence subgroup Γ_{0}(4N) if the following conditions hold:

1) 2|g_{k}, N_{k}|N (k = 1, 2, …, s − 8),  (1.6)
2) 4|N ∑_{k=1}^{s−8} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{s−8} \frac{g_{k}^{2}}{4N_{k}} ,  (1.7)

3) (\frac{∏_{k=1}^{s−8}N_{k}}{|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) (1.8)

for all α and δ with αδ ≡ 1 (mod 4N).

Proof. I. As is well known, the theta-series which contained in the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2) are regular on H. Therefore these functions satisfy condition 1) and by Lemma 1 also condition 4) of the Definition from [1] (p. 53).

II. From (1.6) it follows that

Γ_{0}(4N) ⊂ Γ_{0}(4N_{k}) (k = 1, 2, …, s − 8).  (1.9)

It is easy to verify that (1.7) implies

4|Nδ^{2} ∑_{k=1}^{s−8} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{s−8} \frac{g_{k}^{2}}{4N_{k}} δ^{2φ(2N_{k})−2},  (1.10)

since 2 ∤ δ by virtue of αδ ≡ 1 (mod 4N).

Using (1.9), (1.10) and Lemma 3 from [1] (p. 58), for all substitutions from Γ_{0}(4N) we get:

1) if n = 3, n = 1 and n = 0,

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}) ×

× ∏_{k=3}^{s−8} ϑ_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{k}) =

= i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2}(\frac{∏_{k=1}^{s−8} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

×ϑ′′′_{αg_{r},h_{r}}(τ; 0, 2N_{r})ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t}) ∏_{k=3}^{s−8} ϑ_{αg_{k},h_{k}}(τ; 0, 2N_{k}) (1.11)

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

474 G. LOMADZE

2) if n = 4 and n = 0,

ϑ_{g_{r}h_{r}}^{(4)}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{r})ϑ_{g_{t}h_{t}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{t}) ×

× ∏_{k=3}^{s−8} ϑ_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{k}) =

= i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2}(\frac{∏_{k=1}^{s−8} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

×ϑ_{αg_{r},h_{r}}^{(4)}(τ; 0, 2N_{r})ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t}) ∏_{k=3}^{s−8} ϑ_{αg_{k},h_{k}}(τ; 0, 2N_{k}) (1.12)

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

3) if n = 2 and n = 0,

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}) ×

× ∏_{k=3}^{s−8} ϑ_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{k}) =

= i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2}(\frac{∏_{k=1}^{s−8} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

×ϑ′′_{αg_{1},h_{1}}(τ; 0, 2N_{1})ϑ′′_{αg_{2},h_{2}}(τ; 0, 2N_{2}) ∏_{k=3}^{s−8} ϑ_{αg_{k},h_{k}}(τ; 0, 2N_{k}). (1.13)

Thus by (1.1), (1.11) and (1.2), (1.12), (1.13) we have

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = i^{sη(γ)(sgn δ−1)/2} · i^{(s−8)(1−|δ|)/2} ×

×(\frac{∏_{k=1}^{s−8} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) (j = 1, 2),

which by Lemma 2 implies respectively for j = 1 and j = 2

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = (\frac{∏_{k=1}^{s−8} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l})

if s ≡ 0 (mod 4) and

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) =

= sgn δ(\frac{−1}{|δ|})(\frac{∏_{k=1}^{s−8} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 2 (mod 4).

Hence by (1.5) and (1.8) we obtain

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = χ(δ)(γτ + δ)^{s/2}Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}).

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

III. The functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 1, 2) also satisfy condition 3) of the Definition from [1]. For the proof see [2] (p. 109).

2.

To simplify the notations of this section we put

Ω(τ; g_{1}, h_{1}, c_{1}, N_{1}; g_{2}, h_{2}, h_{2}, c_{2}, N_{2}) =
=\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})  (2.1)

and occasionally

(s − 2)/3 = s̃, (s − 1)/3 = s′.  (2.2)

Lemma 3. For even s ≥ 10 and given N let

Ψ_{3}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = Ω(τ; g_{1}, h_{1}, c_{1}, N_{1}; g_{2}, h_{2}, c_{2}, N_{2}) ×
× ∏_{k=3}^{s/3} ϑ′_{g_{k}h_{k}}(τ; c_{k}, 2N_{k}) if s ≡ 0 (mod 6),  (2.3)
Ψ_{3}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = Ω(τ; g_{1}, h_{1}, c_{1}, N_{1}; g_{2}, h_{2}, c_{2}, N_{2}) ×
× ∏_{k=3}^{s̃} ϑ′_{g_{k}h_{k}}(τ; c_{k}, 2N_{k}) ∏_{k=s̃+1}^{s̃+2} ϑ_{g_{k}h_{k}}(τ; c_{k}, 2N_{k}) if s ≡ 2 (mod 6), (2.4)
Ψ_{3}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = Ω(τ; g_{1}, h_{1}, c_{1}, N_{1}; g_{2}, h_{2}, c_{2}, N_{2}) ×
× ∏_{k=3}^{s′} ϑ′_{g_{k}h_{k}}(τ; c_{k}, 2N_{k})ϑ_{g_{s′+1}h_{s′+1}}(τ; c_{s′+1}, 2N_{s′+1}) if s ≡ 4 (mod 6) (2.5)

476 G. LOMADZE

and

Ψ_{4}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = ∏_{k=1}^{s/3} ϑ′_{g_{k}h_{k}}(τ; c_{k}, 2N_{k}) if s ≡ 0 (mod 6),  (2.6)

Ψ_{4}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = ∏_{k=1}^{s̃} ϑ′_{g_{k}h_{k}}(τ; c_{k}, 2N_{k})ϑ_{g_{s̃+1}h_{s̃+1}}(τ; c_{s̃+1}, 2N_{s̃+1}) ×

×ϑ_{g_{s̃+2}h_{s̃+2}}(τ; c_{s̃+2}, 2N_{s̃+2}) if s ≡ 2 (mod 6),  (2.7)
Ψ_{4}(τ; g_{l}, h_{l}, c_{l}, N_{l}) = ∏_{k=3}^{s′} ϑ′_{g_{k}h_{k}}(τ; c_{k}, 2N_{k})ϑ_{g_{s′+1}h_{s′+1}}(τ; c_{s′+1}, 2N_{s′+1}) (2.8)
if s ≡ 4 (mod 6),

where

2|g_{k}, N_{k}|N (k = 1, 2, …, s/3), 4|N ∑_{k=1}^{s/3} \frac{h_{k}}{N_{k}} if s ≡ 0 (mod 6), (2.9)
2|g_{k}, N_{k}|N (k = 1, 2, …, s̃+ 2), 4|N ∑_{k=1}^{s̃+2} \frac{h_{k}}{N_{k}} if s ≡ 2 (mod 6), (2.10)
2|g_{k}, N_{k}|N (k = 1, 2, …, s′ + 1), 4|N ∑_{k=1}^{s′+1} \frac{h_{k}}{N_{k}} if s ≡ 4 (mod 6), (2.11)

Then we have

(γτ + δ)^{s/2}Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) = ∑_{n=0}^{∞} C_{n}^{(j)}e(\frac{n}{4N} \frac{ατ + β}{γτ + δ} ) (j = 3, 4) (2.12)

for all substitutions from Γ in the neighborhood of each point τ = −\frac{δ}{γ} (γ ≠ 0, (γ, δ) = 1). Proof. I. It is shown in [3] (formula (1.7)) that

(γτ + δ)^{3}Ω(τ; g_{1}, h_{1}, 0, N_{1}; g_{2}, h_{2}, 0, N_{2}) =

= −e(\frac{3}{4} sgn γ)(4N_{1}N_{2}γ^{2})^{−1/2} ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ H_{2} mod 2N_{2} \end{matrix}} ∏_{k=1}^{2} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ω(\frac{ατ + β}{γτ + δ} ; g′_{1}h′_{1}, H_{1}, N_{1}; g′_{2}, h′_{2}, H_{2}, 2N_{2}).  (2.13)

For n = 1 and n = 0 it follows by Lemma 4 from [1] (p. 61) that:

1) (γτ + δ)^{(s−6)/2} ∏_{k=3}^{s/3} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k}) =

= e((s − 6)(sgn γ)/8)(−i sgn γ)^{(s−6)/3} ×

×(2^{(s−6)/3} ∏_{k=3}^{s/3} N_{k}|γ|^{(s−6)/3})^{−1/2} ∑_{\begin{matrix}H_{3} mod 2N_{3} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s/3} mod 2N_{s/3} \end{matrix}} ∏_{k=3}^{s/3} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×ϑ_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) if s ≡ 0 (mod 6),  (2.14)

2) (γτ + δ)^{(s−6)/2} ∏_{k=3}^{s̃} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k})ϑ_{g_{s̃+1}h_{s̃+1}}(τ; 0, 2N_{s̃+1})×

×ϑ_{g_{s̃+2}h_{s̃+2}}(τ; 0, 2N_{s̃+2}) = e((s − 6)(sgn γ)/8) ×

×(2s̃ ∑_{k=3}^{s̃+2} N_{k}|γ|s̃)^{−1/2}(−i sgn γ)^{(s−8)/3} ×

× ∑_{\begin{matrix}H_{3} mod 2N_{3} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s̃+2} mod 2N_{s̃+2} \end{matrix}} ∏_{k=3}^{s̃} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ′_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) ×

× ∏_{k=s̃+1}^{s̃+2} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) (2.15)
if s ≡ 2 (mod 6),

3) (γτ + δ)^{(s−6)/2} ∏_{k=3}^{s′} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k})ϑ_{g_{s′+1}h_{s′+1}}(τ; 0, 2N_{s′+1}) =

= e((s − 6)(sgn γ)/8)(2^{s′−1} ∑_{k=3}^{s′+1} N_{k}|γ|^{s′−1})^{−1/2}(−i sgn γ)^{(s−7)/3} ×

× ∑_{\begin{matrix}H_{3} mod 2N_{3} \\ ... & ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s′+1} mod 2N_{s′+1} \end{matrix}} ∏_{k=3}^{s′} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ′_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k}) ×

×φ_{g′_{s′+1}g_{s′+1}h_{s′+1}}(0, H_{s′+1}; 2N_{s′+1}) ×

×ϑ_{g_{s′+1}h_{s′+1}}(\frac{ατ + β}{γτ + δ} ; H_{s′+1}, 2N_{s′+1}) if s ≡ 4 (mod 6). (2.16)

After multiplying (2.13) successively by (2.14), (2.15) and (2.16), by (2.3) we obtain

1) (γτ + δ)^{s/2}Ψ_{3}(τ; g_{l}, h_{l}, 0, N_{l}) = (−1)^{s/6}e(s(sgn γ)/8)×

×(2^{s/3} ∏_{k=1}^{s/3} N_{k}|γ|^{s/3})^{−1/2} ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s/3} mod 2N_{s/3} \end{matrix}} ∏_{k=3}^{s/3} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ψ_{3}(\frac{ατ + β}{γτ + δ} ; g′_{l}, h′_{l}, H_{l}, N_{l}) if s ≡ 0 (mod 6),  (2.17)

2) (γτ + δ)^{s/2}Ψ_{3}(τ; g_{l}, h_{l}, 0, N_{l}) = (−1)^{(s−2)/6}e(s(sgn γ)/8)×

×(2^{s̃+2} ∏_{k=1}^{s̃+2} N_{k}|γ|^{s̃+2})^{−1/2} ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s̃+2} mod 2N_{s̃+2} \end{matrix}} ∏_{k=3}^{s̃+2} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ψ_{3}(\frac{ατ + β}{γτ + δ} ; g′_{l}, h′_{l}, H_{l}, N_{l}) if s ≡ 2 (mod 6),  (2.18)

3) (γτ + δ)^{s/2}Ψ_{3}(τ; g_{l}, h_{l}, 0, N_{l}) = (−1)^{(s+2)/6}(i sgn γ)e(s(sgn γ)/8)×

×(2^{s′+1} ∏_{k=1}^{s′+1} N_{k}|γ|^{s′+1})^{−1/2} ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s′+1} mod 2N_{s′+1} \end{matrix}} ∏_{k=3}^{s′+1} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ψ_{3}(\frac{ατ + β}{γτ + δ} ; g′_{l}, h′_{l}, H_{l}, N_{l}) if s ≡ 4 (mod 6).  (2.19)

Further, reasoning as in [1] (Lemma 5, the bottom part of p. 64 and p. 65), we obtain (2.12) if j = 3.

II. By Lemma 4 from [1], for n = 1 and n = 0 it respectively follows by (2.6), (2.7) and (2.8) that

(γτ + δ)^{s/2}Ψ_{4}(τ; g_{l}, h_{l}, 0, N_{l}) = (γτ + δ)^{s/2} ∏_{k=1}^{s/3} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k}) =

= e(s(sgn γ)/8)(2^{s/3} ∏_{k=1}^{s/3} N_{k}|γ|^{s/3})^{−1/2}(−i sgn γ)^{s/3} ×

× ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s/3} mod 2N_{s/3} \end{matrix}} ∏_{k=1}^{s/3} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ′_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) =

ON SOME ENTIRE MODULAR FORMS 479

= (−1)^{s/6}e(s(sgn γ)/8)(2^{s/3} ∏_{k=1}^{s/3} N_{k}|γ|^{s/3})^{−1/2} ×

× ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s/3} mod 2N_{s/3} \end{matrix}} ∏_{k=1}^{s/3} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ψ_{4}(\frac{ατ + β}{γτ + δ} ; g′_{l}, h′_{l}, H_{l}, N_{l}) if s ≡ 0 (mod 6),  (2.20)

(γτ + δ)^{s/2}Ψ_{4}(τ; g_{l}, h_{l}, 0, N_{l}) = (γτ + δ)^{s/2} ∏_{k=1}^{s̃} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k}) ×

×ϑ_{g_{s̃+1}h_{s̃+1}}(τ; 0, 2N_{s̃+1})ϑ_{g_{s̃+2}h_{s̃+2}}(τ; 0, 2N_{s̃+2}) =

= e(s(sgn γ)/8)(2^{s̃+2} ∏_{k=1}^{s̃+2} N_{k}|γ|^{s̃+2})^{−1/2}(−i sgn γ)s̃ ×

× ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... \\ Hs̃ mod 2Ns̃ \end{matrix}} ∏_{k=1}^{s̃} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ′_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) ×

× ∑_{\begin{matrix}H_{s̃+1} mod 2N_{s̃+1} \\ H_{s̃+2} mod 2N_{s̃+2} \end{matrix}} ∏_{k=s̃+1}^{s̃+2} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) =

= (−1)^{(s−2)/6}e(s(sgn γ)/8)(2^{s̃+2} ∏_{k=1}^{s̃+2} N_{k}|γ|^{s̃+2})^{−1/2} ×

× ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... & ... \\ H_{s̃+2} mod 2N_{s̃+2} \end{matrix}} ∏_{k=1}^{s̃+2} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ψ_{4}(\frac{ατ + β}{γτ + δ} ; g′_{l}, h′_{l}, H_{l}, N_{l}) if s ≡ 2 (mod 6)  (2.21)

and

(γτ + δ)^{s/2}Ψ_{4}(τ; g_{l}, h_{l}, 0, N_{l}) = (γτ + δ)^{s/2} ∏_{k=1}^{s′} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k}) ×

×ϑ_{g_{s′+1}h_{s′+1}}(τ; 0, 2N_{s′+1})e(s(sgn γ)/8) ×

×(2^{s′+1} ∏_{k=1}^{s′+1} N_{k}|γ|^{s′+1})^{−1/2}(−i sgn γ)^{s′} ×

480 G. LOMADZE

× ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... \\ H_{s′} mod 2N_{s′} \end{matrix}} ∏_{k=1}^{s′} (φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k})ϑ′_{g′_{k}h′_{k}}(\frac{ατ + β}{γτ + δ} ; H_{k}, 2N_{k})) ×

× ∑_{H_{s′+1} mod 2N_{s′+1}} φ_{g′_{s+1}g_{s′+1}h_{s′+1}}(0, H_{s′+1}; 2N_{s′+1}) ×

×ϑ_{g_{s′+1}h_{s′+1}}(\frac{ατ + β}{γτ + δ} ; H_{s′+1}, 2N_{s′+1}) =

= (−1)^{(s+2)/6}(i sgn γ)e(s(sgn γ)/8)(2^{s′+1} ∏_{k=1}^{s′+1} N_{k}|γ|^{s′+1})^{−1/2} ×

× ∑_{\begin{matrix}H_{1} mod 2N_{1} \\ ... & ... & ... & ... & ... & ... & ... \\ H_{s′+1} mod 2N_{s′+1} \end{matrix}} ∏_{k=1}^{s′+1} φ_{g′_{k}g_{k}h_{k}}(0, H_{k}; 2N_{k}) ×

×Ψ_{4}(\frac{ατ + β}{γτ + δ} ; g′_{l}, h′_{l}, H_{l}, N_{l}) if s ≡ 4 (mod 6).  (2.22)

Further, reasoning as in [1] (Lemma 5), we obtain (2.12) if j = 4.

Lemma 4. For even s we have

(a) i^{sη(γ)(sgn δ−1)/2} · i^{(s/3)(1−|δ|)/2} = \begin{cases}1 & if s ≡ 0 (mod 12), \\ sgn δ(\frac{−1}{|δ|}) & if s ≡ 6 (mod 12); \end{cases}

(b) i^{sη(γ)(sgn δ−1)/2} · i^{((s+4)/3)(1−|δ|)/2}=\begin{cases}1 & if s≡8 (mod 12), \\ sgn δ(\frac{−1}{|δ|}) & if s≡2 (mod 12); \end{cases}

(c) i^{sη(γ)(sgn δ−1)/2} · i^{((s+2)/3)(1−|δ|)/2} = \begin{cases}(\frac{−1}{|δ|}) & if s ≡ 4 (mod 12), \\ sgn δ & if s ≡ 10 (mod 12). \end{cases}

Proof. (a) Let s ≡ 0 (mod 6), i.e., s ≡ 0 (mod 12) or s ≡ 6 (mod 12). Then

i^{sη(γ)(sgn δ−1)/2} · i^{(s/3)(1−|δ|)/2} = i^{3hη(γ)(sgn δ−1)} · i^{2h(1−|δ|)/2} =

= \begin{cases}1 & if 2|h, i.e., if s ≡ 0 (mod 12), \\ sgn δ(\frac{−1}{|δ|}) & if 2 ∤ h, i.e., if s ≡ 6 (mod 12). \end{cases}

(b) Let s ≡ 2 (mod 6), i.e., s ≡ 2 (mod 12) or s ≡ 8 (mod 12). Then

i^{sη(γ)(sgn δ−1)/2} · i^{((s+4)/3)(1−|δ|)/2} = i^{(3h+1)η(γ)(sgn δ−1)} · i^{2(h+1)(1−|δ|)/2} =

= i^{(h+1)η(γ)(sgn δ−1)} · (−1)^{(h+1)(1−|δ|)/2} =

= \begin{cases}sgn δ(\frac{−1}{|δ|}) & if 2|h, i.e., if s ≡ 2 (mod 12), \\ 1 & if 2 ∤ h, i.e., if s ≡ 8 (mod 12). \end{cases}

(c) Let s ≡ 4 (mod 6), i.e., s ≡ 4 (mod 12) or s ≡ 10 (mod 12). Then

i^{sη(γ)(sgn δ−1)/2} · i^{((s+2)/3)(1−|δ|)/2} = i^{(3h+2)η(γ)(sgn δ−1)} · i^{2(h+1)(1−|δ|)/2} =

= \begin{cases}(\frac{−1}{|δ|}) & if 2|h, i.e., if s ≡ 4 (mod 12), \\ sgn δ & if 2 ∤ h, i.e., if s ≡ 10 (mod 12). \end{cases}

Theorem 2. For even s and given N the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 3, 4) are entire modular forms of weight \frac{s}{2} and the character

χ(δ) = \begin{cases}(\frac{Δ}{|δ|}) & if s ≡ 0 (mod 4), \\ sgn δ(\frac{−Δ}{|δ|}) & if s ≡ 2 (mod 4) \end{cases}  (2.23)

(Δ is the determinant of the positive quadratic form with integral coefficients in s variables for which the function Ψ_{j} will be used) for the congruence subgroup Γ_{0}(4N) if the following conditions hold:

(a) when s ≡ 0 (mod 6)

1) 2|g_{k}, N_{k}|N (k = 1, 2, …, s/3),  (2.24)
2) 4|N ∑_{k=1}^{s/3} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{s/3} \frac{g_{k}^{2}}{4N_{k}} ,  (2.25)

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

(\frac{∏_{k=1}^{s/3} N_{k}}{|δ|} )Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) = (\frac{Δ}{|δ|})Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}); (2.26)

(b) when s ≡ 2 (mod 6)

1) 2|g_{k}, N_{k}|N (k = 1, 2, …, (s + 4)/3),  (2.27)
2) 4|N ∑_{k=1}^{(s+4)/3} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{(s+4)/3} \frac{g_{k}^{2}}{4N_{k}} ,  (2.28)

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

(\frac{∏_{k=1}^{(s+4)/3} N_{k}}{|δ|} )Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) = (\frac{Δ}{|δ|})Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}); (2.29)

(c) when s ≡ 4 (mod 6)

1) 2|g_{k}, N_{k}|N (k = 1, 2, …, (s + 2)/3),  (2.30)
2) 4|N ∑_{k=1}^{(s+2)/3} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{(s+2)/3} \frac{g_{k}^{2}}{4N_{k}} ,  (2.31)

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

(\frac{∏_{k=1}^{(s+2)/3} N_{k}}{|δ|} )Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) = sgn δ(\frac{−Δ}{|δ|})Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}). (2.32)

Proof. I. As in the case of Theorem 1, the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 3, 4) satisfy condition 1) and by Lemma 3 also condition 4) of the Definition from [1].

II. From (2.24), (2.27) and (2.30) we obtain

Γ_{0}(4N) ⊂ Γ_{0}(4N_{k}).  (2.33)

Obviously, (2.25), (2.28) and (2.31) respectively imply

4|Nδ^{2} ∑_{k=1}^{s/3} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{s/3} \frac{g_{k}^{2}}{4N_{k}} δ^{2φ(2N_{k})−2} if s ≡ 0 (mod 6),  (2.34)

4|Nδ^{2} ∑_{k=1}^{(s+4)/3} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{(s+4)/3} \frac{g_{k}^{2}}{4N_{k}} δ^{2φ(2N_{k})−2} if s ≡ 2 (mod 6), (2.35)

and

4|Nδ^{2} ∑_{k=1}^{(s+2)/3} \frac{h_{k}^{2}}{N_{k}} , 4| ∑_{k=1}^{(s+2)/3} \frac{g_{k}^{2}}{4N_{k}} δ^{2φ(2N_{k})−2} if s ≡ 4 (mod 6). (2.36)

Using (2.33) and Lemma 3 from [1], for all substitutions from Γ_{0}(4N) we obtain:

1) for n = 2, n = 1 and n = 0

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}) ×

ON SOME ENTIRE MODULAR FORMS 483

× ∏_{k=3}^{s/3} ϑ′_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{k}) =

= i^{sη(γ)(sgn δ−1)/2} · i^{(s/3)(1−|δ|)/2}(\frac{∏_{k=1}^{s/3} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

×ϑ′′_{αg_{r},h_{r}}(τ; 0, 2N_{r})ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t}) ×

× ∏_{k=3}^{s/3} ϑ′_{αg_{k},h_{k}}(τ; 0, 2N_{k}) if s ≡ 0 (mod 6),  (2.37)

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}) ×

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

×ϑ_{g_{s̃+1}h_{s̃+1}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{s̃+1})ϑ_{g_{s̃+2}h_{s̃+2}}(\frac{ατ + β}{γτ + δ} ; 0, 2N_{s̃+2}) =

= i^{sη(γ)(sgn δ−1)/2} · i^{(s̃+2)(1−|δ|)/2}(\frac{∏_{k=1}^{s̃+2} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

×ϑ′′_{αg_{r},h_{r}}(τ; 0, 2N_{r})ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t}) ∏_{k=3}^{s̃} ϑ′_{αg_{k},h_{k}}(τ; 0, 2N_{k}) ×

×ϑ_{αg_{s̃+1},h_{s̃+1}}(τ; 0, 2N_{s̃+1})ϑ_{αg_{s̃+2},h_{s̃+2}}(τ; 0, 2N_{s̃+2})  (2.38)
if s ≡ 2 (mod 6),

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}) ×

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

= (sgn δ)i^{sη(γ)(sgn δ−1)/2} · i^{(s′+1)(1−|δ|)/2} ×

×(\frac{∏_{k=1}^{s′} N_{k}}{|δ|} )(γτ + δ)^{s/2}ϑ′′_{αg_{r},h_{r}}(τ; 0, 2N_{r})ϑ_{αg_{t},h_{t}}(τ; 0, 2N_{t}) ×

× ∏_{k=3}^{s′} ϑ_{αg_{k},h_{k}}(τ; 0, 2N_{k})ϑ_{αg_{s′+1},h_{s′+1}}(τ; 0, 2N_{s′+1})  (2.39)
if s ≡ 4 (mod 6),

where r = 2, t = 1 and r = 1, t = 2; 2) for n = 1 and n = 0

∏_{k=1}^{s/3} ϑ′_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} , 0, 2N_{k}) =

= i^{sη(γ)(sgn δ−1)/2} · i^{(s/3)(1−|δ|)/2}(\frac{∏_{k=1}^{s/3} N_{k}}{|δ|} ) ×

×(γτ + δ)^{s/2} ∏_{k=1}^{s/3} ϑ′_{αg_{k},h_{k}}(τ; 0, 2N_{k}) if s ≡ 0 (mod 6), (2.40)

∏_{k=1}^{s̃} ϑ′_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} , 0, 2N_{k})ϑ_{g_{s̃+1}h_{s̃+1}}(\frac{ατ + β}{γτ + δ} , 0, 2N_{s̃+1}) ×

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

×(\frac{∏_{k=1}^{s̃+2} N_{k}}{|δ|} )(γτ + δ)^{s/2} ∏_{k=1}^{s̃} ϑ′_{αg_{k},h_{k}}(τ; 0, 2N_{k}) ×

×ϑ_{αg_{s̃+1},h_{s̃+1}}(τ; 0, 2N_{s̃+1})ϑ_{αg_{s̃+2},h_{s̃+2}}(τ; 0, 2N_{s̃+2})  (2.41)
if s ≡ 2 (mod 6),

∏_{k=1}^{s′} ϑ′_{g_{k}h_{k}}(\frac{ατ + β}{γτ + δ} , 0, 2N_{k})ϑ_{g_{s′+1}h_{s′+1}}(\frac{ατ + β}{γτ + δ} , 0, 2N_{s′+1}) =

= sgn δi^{sη(γ)(sgn δ−1)/2} · i^{(s′+1)(1−|δ|)/2}(\frac{∏_{k=1}^{s′+1} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

× ∏_{k=1}^{s′} ϑ′_{αg_{k}h_{k}}(τ; 0, 2N_{k})ϑ_{αg_{s′+1},h_{s′+1}}(τ; 0, 2N_{s′+1})  (2.42)
if s ≡ 4 (mod 6).

By (2.3), (2.37) and (2.6), (2.40), for j = 3 and j = 4 we get

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = i^{sη(γ)(sgn δ−1)/2} · i^{(s/3)(1−|δ|)/2} ×

×(\frac{∏_{k=1}^{s/3} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 0 (mod 6). (2.43)

By (2.4), (2.38) and (2.7), (2.41), for j = 3 and j = 4 we get

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

×(\frac{∏_{k=1}^{(s+4)/3} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 2 (mod 6). (2.44)

By (2.5), (2.39) and (2.8), (2.42), for j = 3 and j = 4 we get

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = i^{sη(γ)(sgn δ−1)/2} · i^{((s+2)/3)(1−|δ|)/2} ×

×(\frac{∏_{k=1}^{(s+2)/3} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 4 (mod 6). (2.45)

Further, by Lemma 4, for j = 3, 4 realtions (2.43), (2.44), and (2.45) respectively imply:

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = (\frac{∏_{k=1}^{s/3} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l})
if s ≡ 0 (mod 12),

= sgn δ(\frac{−1}{|δ|})(\frac{∏_{k=1}^{s/3} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

× Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 6 (mod 12);

(2.46)

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = sgn δ(\frac{−1}{|δ|})(\frac{∏_{k=1}^{(s+4)/3} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

× Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 2 (mod 12),

= (\frac{∏_{k=1}^{(s+4)/3} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l})

if s ≡ 8 (mod 12); (2.47)

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = sgn δ(\frac{−1}{|δ|})(\frac{∏_{k=1}^{(s+2)/3} N_{k}}{|δ|} )(γτ + δ)^{s/2} ×

× Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l}) if s ≡ 4 (mod 12),

= (\frac{∏_{k=1}^{(s+2)/3} N_{k}}{|δ|} )(γτ + δ)^{s/2}Ψ_{j}(τ; αg_{l}, h_{l}, 0, N_{l})

if s ≡ 10 (mod 12).

(2.48)

From (2.46)–(2.48), (2.26), (2.29), (2.32), and (2.23) it follows that

Ψ_{j}(\frac{ατ + β}{γτ + δ} ; g_{l}, h_{l}, 0, N_{l}) = χ(δ)(γτ + δ)^{s/2}Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 3, 4).

Thus the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 3, 4) satisfy condition 2) of the Definition from [1].

III. By [1] (formula 1.3), for r = 1, t = 2 and r = 2, t = 1 we obtain:

1) If s ≡ 0 (mod 6),

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

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

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

since by (2.24) and (2.25),

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

is a non-negative integer; 2) If s ≡ 2 (mod 6),

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

×ϑ_{g_{s̃+1}h_{s̃+1}}(τ; 0, 2N_{s̃+1})ϑ_{g_{s̃+2}h_{s̃+2}}(τ; 0, 2N_{s̃+2}) =

ON SOME ENTIRE MODULAR FORMS 487

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

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

since by (2.27) and (2.28),

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

is a non-negative integer;

3) If s ≡ 4 (mod 6),

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

× ∏_{k=3}^{s′} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k})ϑ_{g_{s′+1}h_{s′+1}}(τ; 0, 2N_{s′+1}) ×

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

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

since By (2.30) and (2.31)

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

is a non-negative integer.

Analogously,

∏_{k=1}^{s/3} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k}) =

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

×e( ∑_{k=1}^{s/3} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} B_{n}e(nτ) if s ≡ 0 (mod 6),

∏_{k=1}^{s̃} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k})ϑ_{g_{s̃+1}h_{s̃+1}}(τ; 0, 2N_{s̃+1})ϑ_{g_{s̃+2}h_{s̃+2}}(τ; 0, 2N_{s̃+2}) =

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

×e( ∑_{k=1}^{s̃+2} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} C_{n}e(nτ) if s ≡ 2 (mod 6),

∏_{k=1}^{s′} ϑ′_{g_{k}h_{k}}(τ; 0, 2N_{k})ϑ_{g_{s′+1}h_{s′+1}}(τ; 0, 2N_{s′+1}) =

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

×e( ∑_{k=1}^{s′+1} \frac{1}{4N_{k}} (2N_{k}m_{k} + g_{k}/2)^{2}τ) = ∑_{n=0}^{∞} D_{n}e(nτ) if s ≡ 4 (mod 6).

Thus by (2.3)–(2.8) the functions Ψ_{j}(τ; g_{l}, h_{l}, 0, N_{l}) (j = 3, 4) satisfy condition 3) of the Definition from [1].

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.

2. G. Lomadze, On some entire modular forms of half-integral weight for the congruence group Γ_{0}(4N). Proc. A. Razmadze Math. Inst. 118(1998), 101–109.

3. G. Lomadze, Construction of entire modular forms of weights 5 and 6 for the congruence group Γ_{0}(4N). Georgian Math. J. 2(1995), No. 2, 189–199.

(Received 14.04.1998)

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