GENERALIZED RADEMACHER FUNCTIONS AND SOME CONGRUENCE PROPERTIES

Hiroaki NAKAMURA

Department of Mathematics Faculty of Science, Okayama University, Okayama 700-8530, JAPAN

h-naka@math.okayama-u.ac.jp

Abstract In §1, we introduce the classical Dedekind sum and the Rademacher function. Then, in §2, certain generalized Rademacher functions are introduced as the Eichler-Shimura type period integrals of Eisenstein series. In §3, we present a version of continued fraction algorithm which computes efficiently the generalized Rademacher functions. In §4, we show a congruence formula connecting values of the generalized Rademacher functions of weight 2 and weight k > 2. This formula will be applied in a forthcoming paper [N4].

1. The classical Dedekind sums

The classical Dedekind eta function

η(τ) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n})  (q = e^{2πiτ} , τ ∈ ℌ)  (1.1)

on the upper half plane ℌ is one of the most beautiful objects in number theory. Since Δ(τ) = (2πi)^{12}η^{24}(τ) is a cusp form of weight 12 for SL_{2}(ℤ), it is clear that, for any A = \binom{a b}{c d} ∈ SL_{2}(ℤ), the ra-

tio η(Aτ)/\sqrt{\frac{cτ+d}{i}}η(τ) is a 24-th root of unity. The behavior of this little ratio with respect to A ∈ SL_{2}(ℤ) looks delicate and mysteri-

[received: June 18, 2002; accepted in revised form: October 18, 2002]

ous. R. Dedekind, in his note [D] included in Gesammelte Mathematische Werke of B. Riemann, studied the behavior with Riemann’s method. Outstanding is that he found an explicit integer valued function φ : SL_{2}(ℤ) → ℤ controlling this ratio in the form

η(Aτ) = \begin{cases}e^{\frac{2πi}{24} φ(A)}η(τ), & (c = 0); \\ e^{\frac{2πi}{24} φ(A)}\sqrt{\frac{cτ+d}{i}}η(τ), & (c > 0). \end{cases}  (1.2)

Dedekind’s formula for the function φ (which, by definition, factors through PSL_{2}(ℤ) → ℤ) reads as follows:

φ(A) = \begin{cases}\frac{b}{d}, & (c = 0), \\ \frac{a+d}{c} − 12s(a, c), & (c > 0). \end{cases}  (1.3)

Here s(a, c), called now the Dedekind sum, is defined by

s(a, c) = ∑_{i=0}^{c−1} P_{1}(\frac{i}{c})P_{1}(\frac{ai}{c})

with P_{1}(∗) the “sawtooth” function:

P_{1}(x) = \begin{cases}0 & (x ∈ ℤ); \\ x − [x] − \frac{1}{2} & (x ∉ ℤ). \end{cases}

([x]: the greatest integer not exceeding x ∈ ℝ.)

In [R], H. Rademacher intensively studied algebraic properties of the function φ. He derived, for example, the composition formula

φ(AB) = φ(A) + φ(B) − 3sgn(c_{A}c_{B}c_{AB})  (A, B ∈ SL_{2}(ℤ)), (1.4)

where, for any matrix S ∈ SL_{2}(ℤ), c_{S} denotes the lower left entry of S. Following [KM], we shall call φ the Rademacher (φ-)function.

One of the key ingredients that enabled Dedekind to find the above result was to consider the modular transformation of not only η(τ) but also of log η(τ). In fact, Dedekind’s key formula (for the case c > 0) reads:

log η(Aτ) − log η(τ) = −\frac{1}{2} log \frac{i}{cτ + d} + πi(\frac{a + d}{12c} − s(a, c)). (1.5)

This left hand side may also be written in the form:

log η(Aτ) − log η(τ) = −πi ∫_{τ}^{Aτ} E_{2}(z)dz,  (1.6)

where

E_{2}(τ) = \frac{1}{(2πi)^{2}} ∑_{m,n}^{′} \frac{1}{(mτ + n)^{2}}

is the Eisenstein series of weight 2 (with specified conditional convergence). In other words, the Rademacher function φ can be viewed almost as the periods of the Eisenstein series. Unfortunately, E_{2}(τ) is only a quasi-modular form and not a usual modular form. This causes the need of the adjustment term \frac{1}{2} log \frac{i}{cτ+d} of (1.5) which connects φ(A) and the period. To gain a more satisfactory viewpoint, we would need a more subtle treatment by using a non-holomorphic modification of E_{2}. However, we do not enter this subtlety any more here, because this sort of adjustment is rather exceptional phenomenon occurring in this original case among generalizations of E_{2} discussed in the next section.

2. Generalized Rademacher functions

We wish to consider a generalization of the Rademacher function φ introduced in §1 by considering φ as the case of weight 2 and level 1. With regard to the appearance of E_{2} in (1.6), we shall begin by introducing the generalized Eisenstein series E_{k}^{(x_{1},x_{2})}’s (k ≥ 2, (x_{1}, x_{2}) ∈ (\frac{1}{N} ℤ/ℤ)^{2}) given by

E_{k}^{(x_{1},x_{2})}(τ)

:= \frac{(k − 1)!}{(2πi)^{k}} ∑_{a∈(ℤ/Nℤ)^{2}} e^{2πi(x_{1}a_{2}−x_{2}a_{1})} ∑_{m≡a (N)}^{′} \frac{1}{(m_{1}τ + m_{2})^{k}}

= −\frac{P_{k}(x_{1})}{k} + ∑_{\begin{matrix}s∈x_{1}+ℤ \\ s>0 \end{matrix}} ∑_{l=1}^{∞} s^{k−1}e^{2πil(x_{2}+sτ)} + ∑_{\begin{matrix}s∈−x_{1}+ℤ \\ s>0 \end{matrix}} ∑_{l=1}^{∞} s^{k−1}e^{2πil(−x_{2}+sτ)}, (2.1)

where P_{k}(x) : ℝ → ℝ is the k-th periodic Bernoulli function (whose definition will be recalled in (3.4)). It is well known that E_{k}^{(x)} is a holomorphic modular form of level N and weight k, except for the original case E_{2} = E_{2}^{(0)} (cf. [St]).

Let Γ(N) ⊂ SL_{2}(ℤ) be the principal congruence subgroup of level N ≥ 1. For any pair of k ≥ 2 and x = (x_{1}, x_{2}) ∈ (\frac{1}{N} ℤ/ℤ)^{2} with (k, x) ≠ (2, 0), we then consider the Eichler integral of E_{k}^{(x)}:

F_{k}^{(x)}(τ) = − \frac{1}{(k − 2)!} ∫_{τ}^{i∞} (E_{k}^{(x)}(u) + \frac{P_{k}(x_{1})}{k}) (τ − u)^{k−2}du
− \frac{P_{k}(x_{1})}{k} \frac{τ^{k−1}}{(k − 1)!}.  (2.2)

This (indefinite) integral satisfies (\frac{d}{dτ})^{k−1}F_{k}^{(x)} = E_{k}^{(x)}. Writing

j(A, τ) := (cτ + d) for A = \binom{a b}{c d} ∈ Γ(N), we then see that the difference

ϕ_{k}^{(x)}(A)(τ) := j(A, τ)^{k−2}F_{k}^{(x)}(Aτ) − F_{k}^{(x)}(τ)  (2.3)

is a polynomial in τ of degree k−2. What we wish to look closely at here is its “real part” Reϕ_{k}^{(x)}(A)(τ) which is, by definition, the polynomial in the “complex variable” τ whose coefficients consist of the real parts of the corresponding coefficients of ϕ_{k}^{(x)}(A)(τ). It is known that Reϕ_{k}^{(x)}(A)(τ) has only rational coefficients.

The assignment A ↦ Reϕ_{k}^{(x)}(A)(τ) enjoys the 1-cocycle property of the form:

Reϕ_{k}^{(x)}(AB)(τ) = j(B, τ)^{k−2}Reϕ_{k}^{(x)}(A)(Bτ) + Reϕ_{k}^{(x)}(B)(τ) (2.4)

on A, B ∈ Γ(N).

To switch Reϕ_{k}^{(x)}(A)(τ) into the form of 1-cocycle for a left module, let us introduce the space Sym^{k−2}(ℚ^{2}) = ℚ[X, Y]_{deg=k−2} of the homogeneous polynomials in X, Y of degree k − 2 on which SL_{2}(ℤ) acts on the left (written ρ) by

ρ (\begin{matrix}a & b \\ c & d \end{matrix}) . ϕ(X, Y) := ϕ(aX + cY, bX + dY).

Definition (2.5). Let (k, x) ∈ ℤ_{≥2} × (\frac{1}{N} ℤ/ℤ)^{2} with (k, x) ≠ (2, 0). We define the generalized Rademacher function

Φ_{x}^{(k)} : Γ(N) ⟶ Sym^{k−2}(ℚ^{2}) = ℚ[X, Y]_{deg=k−2}

(A = (\begin{matrix}a & b \\ c & d \end{matrix}) ⟼ ∑_{r=0}^{k−2} Φ_{x}^{(r+1,k−1−r)}(A)X^{r}Y^{k−2−r})

by

Φ_{x}^{(k)}(A) := (−1)^{k−1}(k − 2)! X^{k−2} · Reϕ_{k}^{(x)}(A^{−1}) (−\frac{Y}{X} ) .

As expected from the above definition, (2.4) can be converted into the following 1-cocycle property of Φ_{x}^{(k)}:

Φ_{x}^{(k)}(AB) = Φ_{x}^{(k)}(A) + ρ(A).Φ_{x}^{(k)}(B)  (2.6)

on A, B ∈ Γ(N). But one can get a nicer result. In fact, G. Stevens [St] showed how to interpret Φ_{x}^{(k)} as the periods of a certain real differential form extended to the Borel-Serre compactification ℌ̄ of the upper half plane ℌ. With his method, we can extend Φ_{x}^{(k)} canonically to a function from SL_{2}(ℤ) (or even from GL_{2}(ℚ)^{+}) to Sym^{k−2}(ℚ^{2}). Moreover the case of (k, x) = (2, 0) may naturally be included in this unified construction of Φ_{x}^{(k)} with regarding Φ_{0}^{(2)} = −φ(A)/12. Then, the above properties (1.4) and (2.6) can be generalized to

Φ_{x}^{(k)}(AB) = Φ_{x}^{(k)}(A) + ρ(A).Φ_{xA}^{(k)}(B) + \frac{1}{4}δ_{x}^{k=2}sgn(c_{A}c_{B}c_{AB}). (2.7)

Here, sgn(∗) ∈ {±1, 0} denotes the signature of ∗, and the Kronecker symbol δ_{x}^{k=2} is defined by δ_{x}^{k=2} = 1 when (k, x) = (2, 0) and δ_{x}^{k=2} = 0 otherwise.

It will also be nice to have generalizations of the beautiful formula (1.3) to the cases of our generalized Rademacher functions Φ_{x}^{(k)}. Although the author could not find a literature describing the explicit form, by careful calculations one obtains the following formula of Φ_{x}^{(k)} : PSL_{2}(ℤ) → Sym^{k−2}(ℚ^{2}) = ℚ[X, Y]_{deg=k−2};

Φ_{(x_{1},x_{2})}^{(k)} ((\begin{matrix}a & b \\ c & d \end{matrix})) =

\begin{matrix}\{ & \begin{matrix}−\frac{P_{k}(x_{1})}{k} ∫0^{\frac{b}{d}} (tX + Y)^{k−2}dt, & (c = 0); \\ −\frac{P_{k}(x_{1})}{k} ∫0^{\frac{a}{c}} (tX + Y)^{k−2}dt \\ −\frac{P_{k}(ax_{1}+cx_{2})}{k} ∫_{− \frac{d}{c}}^{0} (t(aX + cY) + bX + dY)^{k−2}dt \\ + ∑_{r=0}^{k−2}(−1)^{r}\binom{k−2}{r}X^{r}(aX + cY)^{k−2−r}s_{(x_{1},x_{2})}^{(k−1−r,r+1)}(a, c), & (c > 0). \end{matrix} \\ (2.8) \end{matrix}

where the last factor (called the generalized Dedekind sum) is defined by

s_{(x_{1},x_{2})}^{(k−1−r,r+1)}(a, c) = ∑_{i=0}^{c−1} \frac{P_{k−1−r}(\frac{x_{1}+i}{c})}{k − 1 − r} \frac{P_{r+1}(x_{2} + a\frac{x_{1}+i}{c})}{r + 1}.  (2.9)

The following distribution relation is an important property which connects values of Φ_{x}^{(k)}(A) of different x’s:

Φ_{x}^{(k)}(A) = n^{k−2} ∑_{y∈ \frac{1}{n} x} Φ_{y}^{(k)}(A)  (n ≥ 1).  (2.10)

3. Continued fraction algorithm for Φ_{x}^{(k)}

The calculation of Φ_{x}^{(k)}(\binom{a b}{c d}) by means of (2.8) involve those of gen-

eralized Dedekind sums s_{x}^{(k−1−r,r+1)}(a, c) of (2.9) which need lots of times when c is big. Variants of continued fraction algorithms have been known to improve this sort of inefficiency on computers. We shall exploit a version of it well suited to calculations of our generalized Rademacher functions. The implementation and various numerical tests have been given by Y. Morimoto [Mo] by using PARI-GP and Maple softwares.

Given a matrix A = \binom{a b}{c d} ∈ SL_{2}(ℤ), choose an expression of a/c by a continued fraction of the form:

\begin{aligned}\frac{a}{c} & = k_{0} − \frac{1}{k_{1}−} \frac{1}{k_{2}−} … \frac{1}{−k_{n}} \\ & = k_{0} − \frac{1}{k_{1} − \frac{1}{k_{2} − \frac{1}{^{...} \frac{1}{k_{n−1} − \frac{1}{k_{n}}}}}} \end{aligned}  (3.1)

with an integer sequence (k_{0}, k_{1}, …, k_{n}). This expression may be obtained by first putting r_{0} = a, r_{1} = c > 0 and then defining integers {k_{i}}_{i} with the Euclidean method:

\begin{cases}r_{0} = k_{0}r_{1} − r_{2}, & |r_{1}| > |r_{2}| > 0, \\ r_{1} = k_{1}r_{2} − r_{3}, & |r_{2}| > |r_{3}| > 0, \\ … & … \\ r_{n−1} = k_{n−1}r_{n} − r_{n+1}, & |r_{n}| > |r_{n+1}| = 1, \\ r_{n} = k_{n}r_{n+1} − 0, & |r_{n+1}| = 1, r_{n+2} = 0, \end{cases}  (3.2)

where each k_{i} (i = 0, …, n) is chosen to be either [r_{i}/r_{i+1}] (the greatest integer not exceeding r_{i}/r_{i+1}) or ⌈r_{i}/r_{i+1}⌉ (the least integer greater than or equal to r_{i}/r_{i+1}). Let us write ε = r_{n+1} ∈ {±1}.

Define also a sequence of matrices {A_{i}}_{i} in SL_{2}(ℤ) by

A_{−1} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , A_{0} = (\begin{matrix}k_{0} & −1 \\ 1 & 0 \end{matrix}) , A_{1} = (\begin{matrix}k_{0} & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}k_{1} & −1 \\ 1 & 0 \end{matrix}) ,

…

A_{n} = (\begin{matrix}k_{0} & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}k_{1} & −1 \\ 1 & 0 \end{matrix}) … (\begin{matrix}k_{n} & −1 \\ 1 & 0 \end{matrix}) .

and put

A_{i} = (\begin{matrix}−p_{i} & p_{i−1} \\ −q_{i} & q_{i−1} \end{matrix}) (−1 ≤ i ≤ n).

Then,

Lemma (3.3). Notations being as above, we have

(i) a = −εp_{n}, c = −εq_{n};

(ii) δ := (d − εq_{n−1})/c ∈ ℤ;

(iii) (\begin{matrix}a & b \\ c & d \end{matrix}) = ε (\begin{matrix}k_{0} & −1 \\ 1 & 0 \end{matrix}) … (\begin{matrix}k_{n} & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & δ \\ 0 & 1 \end{matrix}) .

Proof. The Euclidean sequence (3.2) may be rephrased as

\binom{a}{c} = (\begin{matrix}k_{0} & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}k_{1} & −1 \\ 1 & 0 \end{matrix}) … (\begin{matrix}k_{n} & −1 \\ 1 & 0 \end{matrix}) \binom{r_{n+1}}{0}.

The claim (i) follows immediately from this. Both A and εA_{n} are in SL_{2}(ℤ) and have the same left column. Therefore, there exists δ ∈ ℤ with A = εA_{n}\binom{1 δ}{0 1}, i.e.,

(\begin{matrix}a & b \\ c & d \end{matrix}) = ε (\begin{matrix}−p_{n} & p_{n−1} \\ −q_{n} & q_{n−1} \end{matrix}) (\begin{matrix}1 & δ \\ 0 & 1 \end{matrix}) .

The claims (ii) and (iii) follow from this simultaneously.

Definition (3.4). The Bernoulli polynomials B_{k}(X) (k = 0, 1, 2, ...) are defined by the generating function

\frac{te^{tX}}{e^{t} − 1} = ∑_{k=0}^{∞} B_{k}(X)\frac{t^{k}}{k!},

and the Bernoulli number B_{k} is by definition B_{k}(1). We define the periodic Bernoulli function P_{k} : ℝ → ℝ by P_{k}(x) = B_{k}(x − [x]) except for the case k = 1 and x ∈ ℤ where we set P_{1}(n) = 0 (n ∈ ℤ). For x = (x_{1}, x_{2}) ∈ ℚ^{2} and (a, b) ∈ ℤ_{≥0}^{2}, define

β β β_{a,b}(x) := \frac{P_{a}(x_{1})}{max\{a, 1\}} · \frac{P_{b}(x_{2})}{max\{b, 1\}}.

Proposition (3.5) (Continued fraction algorithm). Notations being as above, put k_{n+1} := δ. Then, for x = (x_{1}, x_{2}) ∈ ℚ^{2} and A = \binom{a b}{c d} ∈ SL_{2}(ℤ), we have

Φ_{x}^{(k)}(A) = − ∑_{i=0}^{n} ∑_{j=0}^{k−2}β β β_{j+1,k−1−j}(xA_{i})\binom{k − 2}{j}
× (p_{i−1}X + q_{i−1}Y)^{j}(p_{i}X + q_{i}Y)^{k−2−j}
− ∑_{i=0}^{n+1}β β β_{k,0}(xA_{i−1})

× ∫_{0}^{k_{i}}((−p_{i−1}t + p_{i−2})X + (−q_{i−1}t + q_{i−2})Y )^{k−2}dt
+ \frac{δ_{x}^{k=2}}{4} (sgn(q_{−1}q_{0}) + sgn(q_{0}q_{1}) + ⋯ + sgn(q_{n−1}q_{n})).

Proof. Applying (2.7) iteratedly to (3.3)(iii), we obtain

Φ_{x}^{(k)} = ∑_{i=0}^{n} ρ(A_{i−1}).Φ_{xA_{i−1}}^{(k)}((\begin{matrix}k_{i} & −1 \\ 1 & 0 \end{matrix}))

+ ρ(A_{n}).Φ_{xA_{n}}^{(k)}((\begin{matrix}1 & δ \\ 0 & 1 \end{matrix})) + \frac{δ_{x}^{k=2}}{4} ∑_{i=0}^{n} sgn(q_{i−1}q_{i}).

Then, the formula (2.8) gives

Φ_{xA_{n}}^{(k)}((\begin{matrix}1 & δ \\ 0 & 1 \end{matrix})) = −β β β_{k,0}(xA_{n}) ∫_{0}^{k_{n+1}}(tX + Y)^{k−2}dt,
Φ_{xA_{i−1}}^{(k)}((\begin{matrix}k_{i} & −1 \\ 1 & 0 \end{matrix})) = −β β β_{k,0}(xA_{i−1}) ∫_{0}^{k_{i}}(tX + Y)^{k−2}dt

+ ∑_{j=0}^{k−2}(−1)^{j}\binom{k − 2}{j}
× X^{j}(k_{i}X + Y)^{k−2−j}s_{xA_{i−1}}^{(k−1−j,j+1)}(k_{i}, 1).

Writing here (y_{i}, z_{i}) := xA_{i−1} (i = 0, …, n) so that

(y_{i}, z_{i}) (\begin{matrix}k_{i} & −1 \\ 1 & 0 \end{matrix}) = (y_{i+1}, z_{i+1}) (0 ≤ i < n),

we see that

s_{xA_{i−1}}^{(k−1−j,j+1)}(k_{i}, 1) = \frac{P_{k−1−j}(y_{i})}{k − 1 − j} \frac{P_{j+1}(z_{i} + k_{i}y_{i})}{j + 1}
= (−1)^{k−1−j}β β β_{j+1,k−1−j}(xA_{i}).

Therefore,

Φ_{x}^{(k)} = − ∑_{i=0}^{n+1} ρ(A_{i−1}).β β β_{k,0}(xA_{i−1}) ∫_{0}^{k_{i}}(tX + Y)^{k−2}dt

\begin{aligned}+ & (−1)^{k−1} ∑_{i=0}^{n} ∑_{j=0}^{k−2} ρ(A_{i−1}).\binom{k − 2}{j} \\ & × X^{j}(k_{i}X + Y)^{k−2−j}β β β_{j+1,k−1−j}(xA_{i}) \end{aligned}

+ \frac{δ_{x}^{k=2}}{4} ∑_{i=0}^{n} sgn(q_{i−1}q_{i}).

Noticing that ρ(A_{i−1}) maps

\{ \begin{matrix}X & ⟼ & −p_{i−1}X − q_{i−1}Y, \\ Y & ⟼ & p_{i−2}X + q_{i−2}Y, \\ k_{i}X + Y & ⟼ & −p_{i}X − q_{i}Y, \end{matrix}

we conclude the proof of Proposition (3.5).

4. Some congruence properties

In this section, we shall present a congruence formula (4.3) which connects N^{2} values of Φ_{x}^{(2)}(A) (x ∈ (\frac{1}{N} ℤ/ℤ)^{2}) and the coefficients of Φ_{0}^{(k)}(A). The congruence formula will be applied to the study of a certain measure function on the congruence kernel of SL_{2} which governs the meta-abelian monodromy representation in the fundamental group of the universal elliptic curve minus the origin section ([N1-4]). We refer to the forthcoming paper [N4] as a main reference for details of such an application.

Let ℤ′_{N} ⊂ ℚ denote the ring of fractional numbers with denominators prime to N. For n ≥ 1, write B_{n}^{∗}(X) := B_{n}(X) − B_{n}(0), and define d_{n} (resp. d_{n}^{∗}) to be the least common multiple of the denominators of coefficients of the polynomial \frac{1}{n}B_{n}(X) (resp. \frac{1}{n}B_{n}^{∗}(X)). In the proof of Proposition (4.3) below, we shall frequently make use of the following well known properties:

∑_{x=0}^{N−1} x^{n} = \frac{1}{n + 1}B_{n+1}^{∗}(N) ≡ 0 (mod \frac{N}{d_{n+1}^{∗}} ℤ′_{N}),  (4.1)

\frac{N^{n−1}}{n} (P_{n}(\frac{x}{N} + u) − P_{n}(\frac{x}{N})) ≡ x^{n−1} (P_{1}(\frac{x}{N} + u) − P_{1}(\frac{x}{N}))
(mod \frac{N}{d_{n}} ℤ′_{N})  (4.2)

for x ∈ ℤ, N ∈ ℕ, u ∈ ℤ′_{N}.

Proof. The formula (4.1) is quite popular, and its proof may be left to readers. To prove (4.2), let {t} := t − [t] denote the fractional part of t ∈ ℝ, and put v := \{\frac{x}{N} + u\} − \{\frac{x}{N} \} so that \{u\} = \{v\}. Then, v ∈ ℤ′_{N} and we compute

\frac{N^{n−1}}{n} (P_{n}(\frac{x}{N} + u) − P_{n}(\frac{x}{N}))

= \frac{N^{n−1}}{n} (B_{n}(\{\frac{x}{N} \} + v) − B_{n}(\{\frac{x}{N} \}))

≡ \frac{N^{n−1}}{n} [((\{\frac{x}{N} \} + v)^{n} − \frac{n}{2}(\{\frac{x}{N} \} + v)^{n−1}) − (\{\frac{x}{N} \}^{n} − \frac{n}{2}\{\frac{x}{N} \}^{n−1})]

= \frac{N^{n−1}}{n} [n\{\frac{x}{N} \}^{n−1}v + P_{n−2}(\{\frac{x}{N} \}) + \frac{n}{2}Q_{n−2}(\{\frac{x}{N} \})] .

Here, the congruence is taken modulo \frac{N}{d_{n}} ℤ′_{N}, and P_{n−2}, Q_{n−2} are polynomials of degree n − 2 with coefficients in ℤ. Since \frac{1}{n}B_{n}(X) is a polynomial of the form \frac{1}{n}x^{n} − \frac{1}{2}x^{n−1} + ..., it follows that d_{n} is a multiple of LCM{2, n}. Therefore, the second and third terms of the above last side vanish modulo \frac{N}{d_{n}} ℤ′_{N}. Moreover the integer x′ := \{x/N\}N is congruent to x (mod N). Thus, the above last side continues to

≡ (x′)^{n−1}v ≡ x^{n−1}v = x^{n−1} (P_{1}(\frac{x}{N} + u) − P_{1}(\frac{x}{N})) .

Proposition (4.3). Let N, r, k be integers satisfying N ≥ 1, k ≥ 2 and 0 ≤ r ≤ k − 2, and, for s = 1, ..., k − 2, let e_{s} be the denominator of \frac{B_{k}}{k(k−s)}. Define D_{k,r} to be the least common multiple of the set

\left\{{d_{j+1}d_{k−1−j} (r ≤ j ≤ k − 2), e_{1}, …, e_{k−2} \atop d_{r+1}^{∗}d_{k−r}^{∗}, d_{r+2}^{∗}d_{k−1−r}^{∗}, d_{r+1}^{∗}d_{k−r+1}^{∗}, d_{r+2}^{∗}d_{k−r}^{∗}, d_{r+3}^{∗}d_{k−1−r}^{∗}}\right\}.

Then, for A ∈ Γ(N), we have the following congruence:

12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} \binom{k − 2}{r}x^{k−2−r}(−y)^{r}Φ_{( \frac{x}{N} , \frac{y}{N})}^{(2)}(A)

≡ 12Φ_{0}^{(r+1,k−1−r)}(A) (mod \binom{k − 2}{r}\frac{GCD\{6, N\}N}{D_{k,r}} ℤ′_{N}).

This congruence formula was first proved by the author in the special case of r = 0. The general case of the formula was then conjectured by Y. Morimoto after numerical computations and evidences [Mo].

(4.4) Proof of Proposition 4.3. Let A = \binom{a b}{c d} ∈ Γ(N). Since

\binom{1 gN^{3}}{0 1}\binom{a b}{c d} = \binom{a+cgN^{3} b+dgN^{3}}{c d}, it is sufficient to prove the formula when a ≫ c > 0. In what follows, we shall assume this. Applying (2.7) to the equation \binom{0 −1}{1 0}\binom{c d}{−a −b} = \binom{a b}{c d}, we find

Φ_{(x_{1},x_{2})}^{(k)}(A) = − \frac{P_{k}(x_{2})}{k} ∫_{0}^{− \frac{c}{a}} (tY − X)^{k−2}dt
− \frac{P_{k}(ax_{1} + cx_{2})}{k} ∫_{− \frac{b}{a}}^{0} ((aX + cY)t + bX + dY)^{k−2}dt

+ ∑_{r=0}^{k−2} X^{r}Y^{k−2−r}\binom{k − 2}{r}(−1)^{r} \frac{P_{k−1−r}(x_{1})}{k − 1 − r} \frac{P_{r+1}(x_{2})}{r + 1}

− ∑_{r=0}^{k−2}(aX + cY)^{r}Y^{k−2−r}\binom{k − 2}{r}

× (−1)^{r} ∑_{i=0}^{a−1} \frac{P_{k−1−r}(x_{1} + c\frac{x_{2}+i}{a})}{k − 1 − r} \frac{P_{r+1}(\frac{x_{2}+i}{a})}{r + 1}. (4.4.0)

Let us first evaluate the left hand side of the congruence formula (4.3). When k = 2, r = 0, the above (4.4.0) gives

Φ_{x}^{(2)}(A) = \frac{c}{2a}P_{2}(x_{2}) − \frac{b}{2a}P_{2}(ax_{1} + cx_{2})

+ \{P_{1}(x_{1})P_{1}(x_{2}) − ∑_{i=0}^{a−1} P_{1}(x_{1} + c\frac{x_{2} + i}{a})P_{1}(\frac{x_{2} + i}{a})\} .

Since a ≡ 1, b ≡ c ≡ 0 (mod N), for any integers x, y, P_{2}(\frac{ax+cy}{N}) = P_{2}(\frac{x}{N}). Thus,

12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} \binom{k − 2}{r}x^{k−2−r}(−y)^{r}Φ_{( \frac{x}{N} , \frac{y}{N})}^{(2)}(A)

= \binom{k − 2}{r}(−1)^{r}(S_{0} + S_{1} + S_{2}), (4.4.1)

where

S_{0} := 12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} x^{k−2−r}y^{r}(\frac{c}{2a}P_{2}(\frac{y}{N}) − \frac{b}{2a}P_{2}(\frac{x}{N})),

S_{1} := 12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} x^{k−2−r}y^{r}P_{1}(\frac{x}{N})P_{1}(\frac{y}{N}),

S_{2} := −12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} x^{k−2−r}y^{r}P_{1}(\frac{x}{N} + c \frac{\frac{y}{N} + i}{a})P_{1}( \frac{\frac{y}{N} + i}{a}).

Recalling the notation B_{s}^{∗}(X) := B_{s}(X) − B_{s}(0), by (4.1), we immediately see

S_{0} = \frac{6c}{a} \frac{B_{k−r−1}^{∗}(N)}{k − r − 1} (\frac{B_{r+3}^{∗}(N)}{N^{2}(r + 3)} − \frac{B_{r+2}^{∗}(N)}{N(r + 2)} + \frac{B_{r+1}^{∗}(N)}{6(r + 1)})

− \frac{6b}{a} (\frac{B_{k−r+1}^{∗}(N)}{N^{2}(k − r + 1)} − \frac{B_{k−r}^{∗}(N)}{N(k − r)} + \frac{B_{k−r−1}^{∗}(N)}{6(k − r − 1)}) \frac{B_{r+1}^{∗}(N)}{r + 1}

≡ 0 (mod \frac{GCD\{6, N\}N}{D_{k,r}} ℤ′_{N}).

(Note also here that b ≡ c ≡ 0 (mod N).) Let us rewrite S_{2}. According to the decomposition of P_{1}( \frac{\frac{y}{N} +i}{a}) as \frac{y}{aN} + (\frac{i}{a} − \frac{1}{2}), we may write

S_{2} = S′_{2} + S′′_{2},  (4.4.2)

where

S′_{2} = −12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} x^{k−2−r}y^{r}P_{1}(\frac{x}{N} + c\frac{y + Ni}{aN})(\frac{y}{aN})

= −12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} x^{k−2−r}y^{r}P_{1}(\frac{ax + cy}{N})(\frac{y}{aN})

= −12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} x^{k−2−r}y^{r}P_{1}(\frac{x}{N})(\frac{y}{aN}),

S′′_{2} = −12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} x^{k−2−r}y^{r}P_{1}(\frac{x}{N} + c\frac{y + Ni}{aN})(\frac{i}{a} − \frac{1}{2}).

Noticing that \frac{i}{a} − \frac{1}{2} ∈ \frac{1}{2}ℤ′_{N} and that c\frac{y+iN}{aN} ∈ ℤ′_{N} as a ≡ 1, c ≡ 0 (mod N), we may apply (4.2) to the part “x^{k−2−r}P_{1}(\frac{x}{N} + c\frac{y+iN}{aN})” of S′′_{2} to get the congruence

S′′_{2} ≡ T_{1} + T_{2} (mod \frac{12N}{2d_{k−r−1}} ℤ′_{N})  (4.4.3)

where

T_{1} := − 12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} \frac{N^{k−2−r}}{k − 1 − r}

× (P_{k−1−r}(\frac{x}{N} + c\frac{y + Ni}{aN}) − P_{k−1−r}(\frac{x}{N})) y^{r}(\frac{i}{a} − \frac{1}{2})

= − 12 ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} \frac{1}{k − 1 − r}

× (P_{k−1−r}(c\frac{y + Ni}{a}) − P_{k−1−r}(0)) y^{r}(\frac{i}{a} − \frac{1}{2}),

T_{2} := − 12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} x^{k−2−r}P_{1}(\frac{x}{N})y^{r}(\frac{i}{a} − \frac{1}{2})
= − 12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} x^{k−2−r}P_{1}(\frac{x}{N})y^{r}(−\frac{1}{2}).

388 H. NAKAMURA

Combining the above S_{1}, S′_{2}, T_{2}, we obtain S_{1} + S′_{2} + T_{2}

= 12(∑_{x} x^{k−2−r}P_{1}(\frac{x}{N})) · (∑_{y} y^{r+1})(\frac{1}{N} − \frac{1}{aN})

= 12\frac{a − 1}{aN} \frac{B_{r+2}^{∗}(N)}{r + 2}(\frac{B_{k−r}^{∗}(N)}{N(k − r)} − \frac{B_{k−r−1}^{∗}(N)}{2(k − r − 1)}) ≡ 0 (mod \frac{6N}{D_{k,r}} ℤ′_{N}).

Meanwhile, since

− 12 ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} \frac{1}{k − 1 − r} P_{k−1−r}(c\frac{y + Ni}{a})y^{r}(−\frac{1}{2})

= \frac{6}{a^{k−r}} \frac{P_{k−1−r}(0)}{k − 1 − r} \frac{B_{r+1}^{∗}(N)}{r + 1}

and

−12 ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} −\frac{P_{k−1−r}(0)}{k − 1 − r}y^{r}(\frac{i}{a} − \frac{1}{2}) = −6\frac{P_{k−1−r}(0)}{k − 1 − r} \frac{B_{r+1}^{∗}(N)}{r + 1},

it follows that

T_{1} ≡ −12 ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} \frac{1}{k − 1 − r} P_{k−1−r}(c\frac{y + Ni}{a})y^{r} \frac{i}{a} (mod \frac{6N}{D_{k,r}} ℤ′_{N}).

Summing up the above discussions with (4.4.2-3), we may rewrite (4.4.1) as

12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} \binom{k − 2}{r}x^{k−2−r}(−y)^{r}Φ_{( \frac{x}{N} , \frac{y}{N})}^{(2)}(A)

≡ −12\binom{k − 2}{r}(−1)^{r} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} \frac{1}{k − 1 − r} P_{k−1−r}(c\frac{y + Ni}{a})y^{r} \frac{i}{a} (4.4.4)

modulo \binom{k−2}{r} \frac{GCD\{6,N\}N}{D_{k,r}} ℤ′_{N}. Next, we shall consider the right hand side of the statement formula. Reading the coefficients of X^{r}Y^{k−2−r} of (4.4.0) for (x_{1}, x_{2}) = (0, 0), we get

12Φ_{0}^{(r+1,k−1−r)}(A) = U_{1} + U_{2} + U_{3},

Generalized Rademacher functions and some congruence properties 389

where

U_{1} = − 12\binom{k − 2}{r}\frac{P_{k}(0)}{k}

× {∫_{0}^{−c/a}(−1)^{r}t^{k−2−r}dt + ∫_{−b/a}^{0}(at + b)^{r}(ct + d)^{k−2−r}dt} ,

U_{2} = \frac{12\binom{k−2}{r}(−1)^{r}}{(k − 1 − r)(r + 1)}

× \{P_{k−1−r}(0)P_{r+1}(0) − a^{r} ∑_{i=0}^{a−1} P_{k−1−r}(c\frac{i}{a})P_{r+1}(\frac{i}{a})\} ,

U_{3} = − 12 ∑_{j=r+1}^{k−2} a^{r}c^{j−r}\binom{j}{r}\binom{k − 2}{j}(−1)^{j} ∑_{i=0}^{a−1} \frac{P_{k−1−j}(ci/a)P_{j+1}(i/a)}{(k − 1 − j)(j + 1)}.

As b ≡ c ≡ 0 (mod N), it is easy to see U_{1} ≡ 0 (mod \frac{12\binom{k−2}{r}N}{LCM\{e_{1},...,e_{k−2}\}}ℤ′_{N}). Moreover, taking the equality \binom{j}{r}\binom{k−2}{j} = \binom{k−2}{r}\binom{k−2−r}{j−r} into accounts, we easily see that U_{3} vanishes modulo \frac{12\binom{k−2}{r}N}{LCM\{d_{j+1}d_{k−1−j}|r+1≤j≤k−2\}}ℤ′_{N} (again as c ≡ 0 (mod N)). Now, comparing the distribution relations

\{ \begin{matrix}Φ_{0}^{(r+1,k−1−r)}(A) = N^{k−2} ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} Φ_{( \frac{x}{N} , \frac{y}{N})}^{(r+1,k−1−r)}(A), \\ P_{k}(0) = N^{k−2} ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} P_{k}(\frac{y}{N}) = N^{k−2} ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} P_{k}(\frac{ax + cy}{N}), \end{matrix}

we obtain

U_{2} = \frac{12N^{k−2}\binom{k−2}{r}(−1)^{r}}{(k − 1 − r)(r + 1)} \{∑_{x=0}^{N−1} ∑_{y=0}^{N−1} P_{k−1−r}(\frac{x}{N})P_{r+1}(\frac{y}{N})

− ∑_{x=0}^{N−1} ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} a^{r}P_{k−1−r}(\frac{x}{N} + \frac{cy}{aN} + \frac{ci}{a})P_{r+1}(\frac{i + \frac{y}{N}}{a} )\}

= 12\binom{k − 2}{r}(−1)^{r}\{\frac{P_{k−1−r}(0)}{k − 1 − r} \frac{P_{r+1}(0)}{r + 1}
− ∑_{y=0}^{N−1} ∑_{i=0}^{a−1} a^{r} \frac{P_{k−1−r}(\frac{cy}{a} + \frac{ciN}{a})}{k − 1 − r}N^{r} \frac{P_{r+1}(\frac{i+ \frac{y}{N}}{a} )}{r + 1}\}.  (4.4.5)

We shall decompose the second term of the above last side according to

\frac{(aN)^{r}}{r + 1} P_{r+1}(\frac{y}{aN} + \frac{i}{a}) ≡ \frac{(aN)^{r}}{r + 1} P_{r+1}(\frac{y}{aN}) + y^{r} \frac{i}{a} (mod \frac{N}{d_{r+1}} ℤ′_{N}),

which follows from (4.2) and 0 ≤ \frac{y}{aN} + \frac{i}{a} < 1. We compute:

∑_{i=0}^{a−1} ∑_{y=0}^{N−1} \frac{P_{k−1−r}(\frac{y+iN}{a} c)}{k − 1 − r}(aN)^{r} \frac{P_{r+1}(\frac{y}{aN})}{r + 1}

= ∑_{y=0}^{N−1} a^{−(k−2−r)} \frac{P_{k−1−r}(0)}{k − 1 − r}(aN)^{r} \frac{P_{r+1}(\frac{y}{aN})}{r + 1}

≡ a^{−(k−2−r)} \frac{P_{k−1−r}(0)}{(k − 1 − r)(r + 1)} ∑_{y=0}^{N−1}(\frac{y^{r+1}}{aN} − \frac{r + 1}{2}y^{r})

(mod \frac{N}{d_{r+1}d_{k−r−1}} ℤ′_{N}).

Noticing that the coefficient of X in B_{r+2}(X) is (−1)^{r+1}(r + 2)B_{r+1} and that P_{r+1}(0) = 0 if r + 1 is odd, the above last side continues to

= a^{−(k−2−r)} \frac{P_{k−1−r}(0)}{k − 1 − r}(\frac{B_{r+2}^{∗}(N)}{aN(r + 1)(r + 2)} − \frac{B_{r+1}^{∗}(N)}{2(r + 1)})

≡ \frac{P_{k−1−r}(0)}{k − 1 − r} \frac{(−1)^{r+1}}{r + 1} P_{r+1}(0) (mod \frac{N}{2D_{k,r}} ℤ′_{N})

= \frac{P_{k−1−r}(0)}{k − 1 − r} \frac{P_{r+1}(0)}{r + 1}.

Here we also used a ≡ 1 (mod N). Therefore, we conclude

U_{2} ≡ −\frac{12\binom{k−2}{r}(−1)^{r}}{k − 1 − r} ∑_{i=0}^{a−1} ∑_{y=0}^{N−1} P_{k−1−r}(\frac{y + iN}{a} c)y^{r} \frac{i}{a} (mod \frac{6\binom{k−2}{r}N}{D_{k,r}} ℤ′_{N}),

which coincides with (4.4.4). This completes the proof of Proposition (4.3).

Corollary (4.5). Let k ≥ 2. Then, there exists an integer D(k) (depending only on k) such that for every N ≥ 1,

12 ∑_{x=0}^{N−1} ∑_{y=0}^{N−1}(xY − yX)^{k−2}Φ_{( \frac{x}{N} , \frac{y}{N})}^{(2)}(A) ≡ 12Φ_{0}^{(k)}(A) (mod \frac{N}{D(k)}ℤ′_{N}).

Proof. This is an immediate consequence of Proposition (4.2). We may take D(k) so that

\frac{N}{D(k)}ℤ′_{N} = ∑_{r=0}^{k−2} \binom{k − 2}{r}\frac{GCD\{6, N\}N}{D_{k,r}} ℤ′_{N} ⊂ ℚ.

Example (4.6). Take a matrix A = \binom{122 −4961}{363 −14761} ∈ Γ(11^{2}), and let us examine our congruence formula (4.5) for k = 6, N = 11. In this case, for any 0 ≤ r ≤ 4, \binom{4}{r} \frac{GCD\{6,N\}11}{12D_{6,r}}ℤ′_{11} = 11ℤ′_{11}, hence \frac{N}{12D(6)}ℤ′_{11} may be taken as 11ℤ′_{11} Computation on RHS shows

Φ_{0}^{(6)}(A)(X, Y)

= \frac{6157810527168637}{315}X^{4} + \frac{117260782677249595}{504}Y X^{3}

+ \frac{37381997569467617}{36}Y^{2}X^{2} + \frac{5190578682530622937}{2520}Y^{3}X

+ \frac{1930511018334372017}{1260}Y^{4},

while that on LHS shows

∑_{x,y=0}^{10}(xY − yX)^{4}Φ_{( \frac{x}{11} , \frac{y}{11} )}^{(2)}(A)

= \frac{157339}{6}X^{4} + \frac{557102}{3}Y X^{3} − 102157Y^{2}X^{2}

− \frac{230146}{3}Y^{3}X + \frac{717205}{6}Y^{4}.

In both sides, each coefficient except for that of X^{2}Y^{2} is prime to 11. Now, their difference may be computed as:

Φ_{0}^{(6)}(A)(X, Y) − ∑_{x,y=0}^{10}(xY − yX)^{4}Φ_{( \frac{x}{11} , \frac{y}{11} )}^{(2)}(A)

= \frac{12315621037816679}{630}X^{4} + \frac{117260782583656459}{504}Y X^{3}

+ \frac{37381997573145269}{36}Y^{2}X^{2} + \frac{5190578682723945577}{2520}Y^{3}X

+ \frac{1930511018183758967}{1260}Y^{4}

≡ 0 (mod 11ℤ′_{11}).

Example (4.7). This example was suggested by Y. Morimoto. Let A = \binom{6643 56295}{1539 13042} ∈ Γ(81), and consider the case of weight 10. Then, for 0 ≤ r ≤ 8, \binom{8}{r} \frac{GCD\{6,81\}81}{12D_{10,r}}ℤ′_{3} = 3ℤ′_{3}. The coefficient Φ_{0}^{(5,5)}(A) of X^{4}Y^{4}

in Φ_{0}^{(10)}(A) is computed as

Φ_{0}^{(5,5)}(A) = −\frac{1160735419039913093577749564892899519}{8}

which is prime to 3, while

∑_{x,y=0}^{80} \binom{8}{4}x^{4}(−y)^{4}Φ_{( \frac{x}{81} , \frac{y}{81} )}^{(2)}(A) = −8836456074579123550

which is also prime to 3. Now the difference of these two values is

Φ_{0}^{(5,5)}(A)− ∑_{x,y} \binom{8}{4}x^{4}(−y)^{4}Φ_{( \frac{x}{81} , \frac{y}{81} )}^{(2)}(A)

= −\frac{1160735419039913022886100968259911119}{8}

which is divisible by 3 (but not divisible by 9).

Example (4.8). Let us consider the case of weight 4 for A = \binom{385 15616}{15744 638593} ∈ Γ(128). Then, \binom{2}{r} \frac{GCD\{6,128\}128}{12D_{6,r}}ℤ′_{2} is 4ℤ′_{2} for r = 0, 2, and 8ℤ′_{2} for r = 1. The computation shows

Φ_{0}^{(4)}(A) = \frac{1236394022}{45}X^{2} + \frac{33706991732}{15}XY + \frac{229732849464}{5}Y^{2},

and

∑_{x,y=0}^{127}(xY − yX)^{2}Φ_{( \frac{x}{128} , \frac{y}{128} )}^{(2)}(A)

= \frac{80854186}{3}X^{2} − \frac{10808380}{3}XY − \frac{80165720}{3}Y^{2}.

Both of them do not belong to 4ℤ′_{2}[X, Y] (because of the coefficients of X^{2}). Now the difference of these two polynomials is computed as

Φ_{0}^{(4)}(A) − ∑_{x,y=0}^{127}(xY − yX)^{2}Φ_{( \frac{x}{128} , \frac{y}{128} )}^{(2)}(A)

= \frac{23581232}{45}X^{2} + \frac{33761033632}{15}XY + \frac{689599376992}{15}Y^{2}

Generalized Rademacher functions and some congruence properties 393

which belongs to 4ℤ′_{2}[X, Y] (actually to 16ℤ′_{2}[X, Y]).

At the time of writing this paper, the author does not have an example assuring whether our estimate of the modulus is best possible with respect to 2-powers.

Acknowledgements

The author would like to thank Yasuhiko Morimoto for valuable discussions on the topic and useful calculations of numerical examples. He is also very grateful to the referee for crucial comments which were very helpful to improve the submitted version of this paper.

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