数理解析研究所講究録1052 巻 1998 年 201-216

201

A remark on Serre’s example of p–adic Eisenstein series
byS. NAGAOKA
1$($$\mathrm{L}\backslash *\not\in_{\mathrm{R}}^{\mathrm{f}}\cdot\kappa(^{\vee}\sim;\pm \mathrm{E}-\mathrm{r}\cdot\not\in[6]\mathrm{a}_{5}^{\tau})$Introduction.

In [Se], J. P. Serre developed the theory of padic modular forms and applied
it to the construction of p–adic zeta function. In this paper, we shall try to
generalize a formula for p–adic Eisenstein series which was originally given by
Serre. A p–adic modular form is a formal power series$f= \sum_{t=0}^{\infty}a(t)q^{t}\in \mathbb{Q}p[[q]]$which is the limit of a sequence of modular forms
coefficients: $\lim_{marrow\infty}f_{m}=f$ .
If we denote by$\{f_{m}\}$with rational Fourier$f_{m}= \sum_{=t0}^{\infty}a((m)t)q^{t}\in \mathbb{Q}[[q]]$the Fourier expansion of$f_{m}$( -expansion), this limit means that$q$$v_{p}(f-fm)$ $:= \inf_{t}v_{p}(a(t)-a^{(}(m)t))arrow+\infty$$(marrow\infty)$

,

where
is the valuation of
normalized as $v_{p}(p)=1$ . If we denote by
the weight of, then Serre showed thathas the limit in the followingset:$v_{p}$$\{k_{m}\}$$\mathbb{Q}_{p}$$\{f_{m}\}$$\{k_{m}\}$$X:=\varliminf X/(p-1)p^{m}-1\mathbb{Z}=\mathbb{Z}\cross p\mathbb{Z}/(p-1)\mathbb{Z}$Let
be the Siegel-Eisenstein series of degree
definition, see \S 2). Set$E_{k}^{(n)}$$G_{k}$

where$:= \frac{1}{2}\zeta(1-k)E_{k}(1)$$n$.

and weight$k$(for precise,

is the Riemann zeta function. For $k\in X$ , we take a sequencesuch that $\lim_{marrow\infty}k_{m}=k$ and $|k_{m}|arrow+\infty(marrow\infty)$ . Serredefined the p–adic Eisenstein series
of weight $k\in X$ by$\zeta(s)$$\{k_{m}\}\subset 2\mathbb{Z}$$G_{k}^{*}$$G_{k}^{*}$$:= \lim_{marrow\infty}c_{k_{m}}$.

The right-hand side converges a.nd it becomes a p–adic modular form. The
following example is due to Serre:

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EXAMPLE of.

let $p>3$ be a$k=(1, R \frac{+1}{2})\in X.$ Then we have
$G_{k}^{*}$$p_{7\dot{2}}me$number such that

$G_{k}^{*}=h(-_{\mathrm{P}})+ \sum^{\infty}\sum_{tt=10<d|}(\frac{d}{p})q^{t}$where $h(-p)$ is the class number of the quadratic field(mod 4) and$p\equiv 3$,$\mathbb{Q}(\sqrt{-p})$.

The main purpose of this paper is toa generalization of this example. The
Siegel modular form $f(Z)$ has a Fourier expansion. of the form$\circ\sigma \mathrm{i}\mathrm{v}\mathrm{e}$$f(Z)= \sum_{T}a_{f}(T)\exp[2\pi\sqrt{-1}\mathrm{t}\mathrm{r}(TZ)]=\sum af(\tau)q\tau\tau$,where runs over the set of, positive semi-definite symmetric matrices (see ). For $T=(t_{ij})$ and $Z=(z_{ij})$ , we set $q_{ij}:=\exp(2\pi\sqrt{-1}Zij),$ $qi=qii$ ,and. Then can be.
as a power series in
So we can define the p–adic Siegel modular form as an element of. Our result can be stated as follows:$T$$\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{f}- \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}1$$\mathrm{S}.2$$t_{i}=t_{ii}$$f$$\mathbb{C}[q_{ij}, q_{ij}-1][[q_{1}, \ldots, q_{n}]]$$\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}\cdot \mathrm{d}\mathrm{e}\mathrm{d}$$\mathbb{Q}[qij, q^{-1}ij]$$[[q_{1}, \ldots , q_{n}]]$THEOREM Let $p>3$ be a$\mathrm{P}^{7\dot{\eta}me}$$k_{m}$

then the sequence$E_{k}^{*}$$:=$$\{k_{m}\}$number such that$p\equiv 3$$:=1+ \frac{p-1}{2}\cdot pm-1\in \mathbb{Z}$(mod 4). If we put,

has the limit $k–(1, L^{+\underline{1}})2\in X$ and$\lim_{marrow\infty}(^{\frac{1}{2}\zeta(1}-k_{m})E_{k)}m(2)$$=$$\frac{1}{2}h(-p)+D(T)=\tau\geq 0\sum_{0-p_{\mathit{0}\Gamma}}$rank$(T) \sum_{<0d1^{\epsilon()}\tau}(\frac{d}{p})q^{T}$,

where $D(T)$ is the discriminant of the field $\mathbb{Q}(\sqrt{-\det(2\tau)})$ and we understand$D(T)=0$ if $\det(\tau)=0_{f}$ and$\epsilon(T):=\mathrm{g}.\mathrm{c}.\mathrm{d}(t_{11},2t_{12}, t22)$.

In the final section, we give an additional formula which is concerned with
reduction mod of the Fourier coefficient of the Siegel-Eisenstein series.
$p$2

Siegel-Eisenstein series.

Let$\mathbb{H}_{n}$be the Siegel upper half space of degree :$n$$\mathbb{H}_{n}:=\{Z=X+\sqrt{-1}Y\in \mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{C})|Y>0\}$

The real symplectic group$\mathrm{S}_{\mathrm{P}_{n}}(\mathbb{R})$acts on$\mathbb{H}_{n}$$Z\mapsto M\langle Z\rangle:=(AZ+B)(Cz+D)^{-1}$ ,.

by$M=\in \mathrm{S}_{\mathrm{P}_{n}}(\mathbb{R})$.

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The group
denote theAny element inis called theSie-gel modular group. Let.
space-vectorof Siegel modular forms of weight for.
admits a Fourier expansion of the form$\Gamma_{n}:=\mathrm{S}_{\mathrm{P}_{n}}(\mathbb{R})\cap M_{2n}(\mathbb{Z})$$[\Gamma_{n}, k]$

$f$

(2.1)

$\mathbb{C}$$f(Z \mathrm{I}=\sum_{\Lambda_{n}0\leq\tau\in}af(\tau)\exp[2\pi\sqrt{-1}\mathrm{t}\mathrm{r}(Tz)]$

where the index set
(2.2)

Let

$k$

$\Lambda_{n}$,is defined by.

$\Lambda_{n}:=\{T=(i_{ij})\in \mathrm{S}\mathrm{y}\mathrm{m}_{n}(\mathbb{Q})|t_{ii}\in.\mathbb{Z}, 2t_{ij}\in \mathbb{Z}\}$$\Gamma_{n,0}$$\Gamma_{n}$$[\Gamma_{n}, k]$

be the subgroup of$\Gamma_{n}$defined by$:=\{\in\Gamma_{n}|C=O_{n}\}$ .

$\Gamma_{n,0}$

For an even integer , we define a series
$k$

(2.3)

$E_{k}^{(n)}(Z):=$$\sum$($c*$$\det(Cz+D)^{-k}$ ,$Z\in \mathbb{H}_{n}$.

$D*)\in 1_{n,0\backslash }’\Gamma_{n}$This series is absolutely convergent if $k>n+1$ and it becomes a Siegel modular
form of weight for:. Here we call this the Siegel-Eisensteinseries of degree and weight . We write the Fourier expansion ofby$k$$\Gamma_{n}$$E_{k}^{(n)}\in[\Gamma_{n}, k]$(2.4)$E_{k}^{(n)}$

$k$

$n$

$E_{k}^{(n)}(z)= \sum_{\Lambda_{n}0\leq\tau\in}.a^{()}kn(\tau)\exp[2\pi..\sqrt{-1}\mathrm{t}\mathrm{r}(Tz)]$.

It is known that any Fourier coefficientis rational ([Si]). The explicit
formula ofwas studied by several authors ([Kau], [M], [Kat]). For
later purpose, we shall introduce an abbreviation. For $T=(t_{ij})\in\Lambda_{n}$ and, we write$a_{k}^{(n)}(T)$$a_{k}^{(n)}(T)$$Z=(z_{ij})\in \mathbb{H}_{n}$(2.5)

where.

$q^{T}:= \exp[2\pi\sqrt{-1}\mathrm{t}\mathrm{r}(\tau z)]=\prod_{i<j}q^{2t}ij\prod_{1}^{n}jq^{t:}i=i$$q_{ij}:=\exp(2\pi\sqrt{-1}z_{ij})$(2.1) can be rewritten as’, and $q_{i}=q_{ii},$ $t_{i}=t_{ii}$ . So the Fourier expansion$f= \sum_{\Lambda_{n}0\leq T\in}af(T)q\in \mathbb{C}\tau[q_{ij}, qij1-][[q_{1}, \ldots, q_{n}]]$namely,$f$

is regarded as an.

$[[q_{1}, \ldots, q_{n}]]$,elem.e.nt of the formal power series ring$\mathbb{C}[qij, q_{ij}-1]$

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3

Bernoulli numbers and generalized Bernoulli numbers.

In this section we review some of the basic facts about Bernoulli numbers andgeneralized Bernoulli numbers. The ordinary Bemoulli numbers
are definedby$B_{m}$(3. 1)$\frac{t}{e^{t}-1}=\sum_{m=0}^{\infty}B_{m^{\frac{t^{m}}{m!}}}$.

As is well known, certain special values of the Riemann zeta function can be
represented by the Bernoulli numbers: for any eveh positive integer $m$ , we have
(3.2)$\zeta(1-m)=-\frac{B_{m}}{m}$THEOREM 3.1 (1) (Kummer) If $m$ and(mod $p^{e-1}(p-1)$ ) and(3.3)$n\not\equiv \mathrm{O}$.

are positive even integers with $m\equiv n$(mod $p-1$ ), then
$n$

(mod

$(1-p^{m-1}) \frac{B_{m}}{m}\equiv(1-p^{n-1})\frac{B_{n}}{n}$$p^{e}$).

(cf. $[W/,$, Corollary 5.14).
(2) (von Staudt- Clausen) Let $m$ be even and positive. Then$\mathrm{S}.\mathit{5}.\mathit{3}$

(3.4)

.

$B_{m}+ \sum_{\dagger?\sim||\mathrm{n}}\frac{1}{p}\in \mathbb{Z}$Consequently, $pB_{m}$ is -integral for all $m$ and all(3) (Carl,itz) If $p^{e-1}(p-1)|m$ , then we have
$p$

(35)

$pB_{m}\equiv p-1$

(cf.
(4) Let $p>3$ be a

(mod

$p$

$p^{e}$. (cf. $[W]$, Theorem 5.10).

).

).

$[W/fp.\mathit{8}\mathit{6},\mathit{5}.\mathit{1}\mathit{1}(b)$$p_{7\dot{\eta}}me$(3.6)

number such that$p\equiv 3$$B_{2_{\frac{+1}{2}}} \equiv-\frac{h(-p)}{2}\not\equiv 0$(mod 4). Then we have(mod ).
$p$(cf. $[BS/$, Chap.5, \S 8, Problem 4 and $[WJ$, p.86, Exercise 5.9).

Let be a Dirichlet character of conductor $f=f_{\chi}$ . The generalized Bemoulli
numbers
are defined by$\chi$$B_{m,\chi}$(3.7)$\sum_{a=1}^{f}\frac{\chi(a)te^{at}}{e^{ft}-1}=\sum_{m=0}^{\infty}B_{m,x}\frac{t^{m}}{m!}$.

Note that( : the principal character) except for $m=1$ , where
we have.
Let
be the Dirichlet -function
to a Dirichlet character :$B_{m,\chi^{\mathrm{O}}}=B_{m}$$B_{1,\chi^{0}}= \frac{1}{2},$$L(\mathit{8};\chi)$$\chi^{\mathrm{O}}$$B_{1}=- \frac{1}{2}$$L$$\mathrm{b}\mathrm{e}1_{\mathrm{o}\mathrm{n}}\mathrm{g}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\chi$(3.8)$L(s;x)$$:= \sum_{m=1}^{\infty}\frac{\chi(m)}{m^{s}}$.

205
Then, for any$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{r}m\geq 1$(3.9), we have$L(1-m; \chi)=-\frac{B_{m,\chi}}{m}$( . cf. [I], 8.2, Theorem 1). In the following, we shallstate Carlitz’s result
about generalized Bernoulli numbers in the case that is quadratic.
$\mathrm{e}_{\mathrm{o}}.\sigma$$\chi$

THEOREM 3.2 (Carlitz [Ca]) Suppose that is a quadratic
character of conductor .
(1) Ifthenis a rational inieger for every $m\geq 0$ and ifisnot a power of a
, then evenis a rational integer.
(2) If is a rationalsuch that $p^{e}|m$ but \dag er . , thendivides thenumerator of
If is divisible by at least two
and is arbitrary$Di7\dot{\eta}Chlet$$\chi$$f_{\chi}$$\chi\neq x^{0_{J}}$$f_{\chi}B_{m,\chi}$$f_{\chi}$$p_{7\dot{\eta}}me$$p$$\frac{1}{m}B_{m,\chi}$$p_{7\dot{\eta}}me$$p$$B_{m,\chi},\cdot$$p^{e}$$f_{\chi}$$f_{\chi}$$p_{7}\dot{\mathrm{v}}meS$$p(\mathit{3})SuppoSethatf_{x^{=}pp}pisanodd7\dot{\eta}m7\dot{\mathrm{V}}me_{J}thenagainpdivideSthenumera_{J}torofeB-^{\mathrm{i}’}eand\mathrm{e}||7\gamma 1mx\cdot$(3.10)(mod$pB_{m,\chi}\equiv p-1$

$p$

. Then$p^{e}\rangle$

if $j(p-1)=2m$ for some odd .
$j$in$\mathbb{Q}(\chi)$defined by$\wp=(pJ^{\cdot}1-\chi(g)g)\gamma\uparrow 1$

where

$g$

is a primitive root mod . If $\wp\neq(1),$$p$

In our case, is quadratic, namely,
some odd , then$\chi$$\mathrm{t}1_{1}\mathrm{e}\iota \mathrm{l}$(mod$pB_{m,\chi}\equiv p-1$,$\mathbb{Q}(\chi)=\mathbb{Q}$.

).

$\wp^{e}$$\mathrm{O}\mathrm{l}$), if $j(p-1)=2m$ for$\mathrm{V}\mathrm{i}\mathrm{o}\iota \mathrm{l}\mathrm{S}\mathrm{l}\mathrm{y}$$j$$\chi(g)g^{m}\equiv 1$(mod p).

Therefore, Theorem 3.2, (3) is a special case of Carlitz’s result.
4

Fourier coefficients of Siegel-Eisenstein series.

In this section, we shall introduce some explicit formulas of Fourier coefficientof).
-Eisenstein series in the case
It is well known thatis:$a_{k}^{(n)}(T)$$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{o}\sigma \mathrm{e}\mathrm{l}$$n\leq\cdot.$$a_{k}^{(1)}(t)(4\leq k\in 2\mathrm{Z})$(4.1)$\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{c}\iota 1\mathrm{e}\gamma \mathrm{s}\mathrm{f}\mathrm{o}\downarrow \mathrm{l}\mathrm{o}\mathrm{w}\mathrm{s}$$- \frac{2k}{B_{k}}\sigma_{k-1}(t)$$\mathrm{i}(^{\backslash }$$a_{k}^{(1)}(t)=\{$

1

if

where
.
In the case $n=2$ , G. Kaufllold [Kau] and H.
Here we introduce a description ofby M.

$t>0$ ,
$t=0$ ,$\sigma_{m}(t):=\sum_{0<d|t}ff^{\mathrm{t}}$gave explicit formulas.
. and D. Zagier [EZ] in$\mathrm{M}_{\mathrm{c}}^{i}\mathrm{k}\gamma \mathrm{S}\mathrm{S}[\mathrm{M}|$$a_{k}^{(2)}$$\mathrm{E}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{l}\mathrm{l}\mathrm{G}\iota$

206
which they used Cohen’s function $H(r, N)$ .
Let and $N$ be non negative integers with $r\geq 1$ . For $N\geq 1$ , we define$r$$(-1)^{[\frac{f}{2}]_{(}}r-1)!N^{r-\frac{1}{2}}21-r\pi-r_{L}(r;\chi(-1)’\backslash xr)$$h(r, N):=\{$$0$where $L(s;x)$ is the Dirichlet. Moreover, for$\chi_{D}(d)=(\frac{D}{d})$

$L$

if

$(-1)^{r}N\equiv 0$

orl

if

$(-1)^{r}N\equiv 2$or3 (mod 4) ,-function and we write, we define$\chi_{D}$(mod 4),for the character$*$$N\in \mathbb{R}$$\cdot$$\sum_{d^{2}|N}h(r,$if$f \frac{N}{f})$$H(r, \mathit{1}\mathrm{V}):=\{$$(-1)^{r}N\equiv 0\mathrm{o}\mathrm{r}.1$$\zeta(1-2r)$

if $N=0$ ,$0$otherwise.

$($

mod

$4)_{:}.’\backslash ^{-}’>0.$,.

The above defined function $H(r, N)$ is called Cohen ノ s $functi_{on}$ It is.
$H(r, N)$ has the following description.
LEMMA 4.1 (, p.273, ) If we set
vrith $D$ discriminantof a quadratic field, then we have$\mathrm{k}\mathrm{n}\mathrm{o}\iota \mathrm{V}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{a}^{\wedge}$$\cdot$$[\mathrm{c}_{0}]$(4.2)$(-1)^{r}\mathit{1}\mathrm{V}=Df^{2}$$)$

Returning to the$WI\dot{o}$bius function.

$\mathrm{f}\mathrm{o}\mathrm{r}\mathfrak{m}\mathrm{u}\mathrm{l}\mathrm{a}a_{k}^{(}(2)\tau)$, for$\mathit{0}_{2}\neq T\in\Lambda_{2}$(cf. (2.2))., we define$\epsilon(T):=\max\{l\in \mathbb{N}|l-1T\in\Lambda 2\}$

THEOREM 4.2 ([EZ], p.80, Corollary 2) If
(4.4)

:

$H(r, N)=L(1-r; \chi_{D})\sum_{<0d|f}\mu(^{r}d)\chi_{D}(d)d^{\Gamma-1}\sigma_{2_{\Gamma-}1}(\frac{f}{d})$where $\mu(d)$ is the(4.3)$\mathrm{c}$.

$0\leq T\in\Lambda_{2}(\mathcal{I}’\neq O_{2})$$a_{k}^{(2)}.(T)= \frac{4k(k-1)}{B_{k}\cdot B_{2k}-2}\sum_{0<d|\mathcal{E}(\tau)}d^{k1}-H(k-1,$$\frac{\det(2T)}{d^{2}})$. then.

Especially, if rank $T=1$ , then(4.5)

$a_{k}^{(2)}(T)=- \frac{2k}{B_{k}}$.

$0| \sum_{<d\xi(T)}d^{k}-1=-\frac{2k}{B_{k}}\sigma_{k1}-(\epsilon(T))$.

REMARK. It should be noted that the factor $4k(k-1)/B_{k}\cdot B_{2k-2}$ in (4.4) is
missing in the original formula of Eichler and Zagier.
By using (4.2), we can rewrite the formula (4.4). For(4.6)$0<T\in\Lambda_{2},\cdot$

we$\backslash \mathrm{v}.\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{e}$$-\det(2\tau)=D(T)\cdot f(T)^{2}$ ,where $D(T)$ is the discriminant of the imaginary quadratic field $\mathbb{Q}(\sqrt{-\det(2T)})$and $f(T)\in$ N. It is quite obvious that the number $f(T)$ is divisible by:$\epsilon(T)|f(T)$ .
$\epsilon(T)$207
COROLLARY 4.3 (Explicit formula of$a_{k}^{(2)}( \tau)=-.\frac{4k\cdot B_{k-1}x_{D}(T)}{B_{k}\cdot B_{2k}-2},F_{k}(T)$$a_{k}^{(2)}(T)$) For $0<T\in\Lambda_{2}$ , we have

,

(4.7)
$F_{k}(T)= \sum dk-1\sum_{T0<d|\epsilon(T)0<f|\angle \mathrm{L}\mathit{1}d}\mu(f)\chi_{D}(T)(f)f^{k}-2\sigma_{2k}-3(\frac{f(T)}{fd})$.

5$p\cdot \mathrm{a}\mathrm{d}\mathrm{i}\mathrm{c}$.

Eisenstein series.

As we mentioned in Introduction, J. P. Serre developed the theory of p-adic
modular form and applied it to the construction of
zeta function. The
p-adic Eisenstein series is a typical example of p–adic modular form. In thissection, we shall briefly review Serre’s theory.
In the following, for simplicity, we assume that is an odd prime. Put$p- \mathrm{a}\mathrm{d}\backslash \mathrm{i}_{\mathrm{C}}$$p$$X_{m}$Then$\{X_{m}\}$

(5.1)

,

$:=\mathbb{Z}/p^{m-1}(_{\mathrm{P}}-1)\mathbb{Z}=\mathbb{Z}/p^{m-1}\mathbb{Z}\cross \mathbb{Z}/(p-1)\mathbb{Z}$$m\geq 1$.

forms a projective system. Let $X$ be the limit of this system:$X:=\varliminf x_{m}=\mathbb{Z}_{p}\cross \mathbb{Z}/(p-1)\mathbb{Z}$

,

where
is the ring of p–adic integers.
The -adic modular form$\mathbb{Z}_{\mathrm{p}}$

$p$

(5.2)

$f= \sum_{t=0}^{\infty}a(t)qt\in \mathbb{Q}_{p}[[q]]$is defined as the limit of a sequence of modular forms
wvith rational Fourier(thecoefficients. The limit means the following. Let
be the valuation on
field of p–adic numbers) normalized as $v_{p}(p)=1$ . We denote by$\{f_{m}\}$$\mathbb{Q}_{p}$$v_{p}$$f_{m}= \sum_{t=0}^{\infty}a^{(m)}(t)q^{t}\in \mathbb{Q}[[q]]$the Fourier expansion of$f_{m}$. The$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}_{\mathrm{o}}^{\sigma}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{m}marrow\infty f_{m}=f$means that$(marrow\infty)$$v_{p}(f-f_{m}):= \inf_{t}v_{p}(a(t)-a((m)t))arrow+\infty$.

We denote byhas. Serre [Se] showed thatthe weight of$X$the limit in . This element $k\in X$ is called the weight of p–adic modular
form . The p–adic Eisenstein series.(in the sense of Serre) is defined as follows.
Put$\{k_{m}\}\subset 2\mathbb{Z}$$\{k_{m}\}$$\{f_{m}\}$

$k$

$f$

$G_{k}:= \frac{1}{2}\zeta(1-k)E^{(1)}k-\frac{B_{k}}{2k}E^{(1)}k=$

,

208
where
By (4.1),$E_{k}^{(1)}$is the-Eisenstein series of degree 1 and weight $k(4\leq k\in 2\mathbb{Z})$ .
has a Fourier expansion of the form$\mathrm{S}\mathrm{i}\mathrm{e}_{\epsilon}\sigma,\mathrm{e}1$$G_{k}$$G_{k}=- \frac{B_{k}}{2k}+\sum_{t=1}^{\infty}\sigma_{k}-1(t)q^{t}\in \mathbb{Q}[[q]]$Assume that $k\in X$ . For an.

$*$, we can define a p-adic$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{r}t\geq 1$by$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{r}\sigma^{*}-(k1t)$.
$\sigma_{k-1(}^{*}t):=\sum_{1(d,p)=}d^{k-1}0<d|t’$

If $k\in X$ is even, then we can choose a sequence ofsuch that $k_{m}arrow k\in X$ and $|k_{m}|arrow+\infty$ where
value. For this, we have$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{r}\mathrm{S}\{k_{m}\}(4\leq k_{m}\in 2\mathbb{Z})$is the ordinary absolute$|\cdot|$$\{k_{m}\}$$\lim_{marrow\infty}\sigma_{k_{m^{-1}}}(t)=\sigma_{k-}*1(t)$in . The -adic Eisenstein se
defined by$\mathbb{Z}_{p}$

$p$

(5.3)

.(of degree 1) and weight $k\in X-\{0\}$ is$7\dot{\mathrm{v}}es$$G^{*}k= \lim_{marrow\infty}ck_{m}$.

Namely,(5.4),$G_{k}^{*}= \frac{1}{2}\zeta^{*}(1-k)+\sum_{t=1}\sigma^{*}-1(kt\infty)q.\in \mathbb{Q}_{p}t[[q]]$where the convergence of the constant term is guaranteed in [Se], 1.5, Cor. 2,
andis essentially the p–adic zeta function of Kubota and Leopoldt. Strictly
speaking, if $(s, u)\in X=\mathbb{Z}_{p}\cross \mathbb{Z}/(p-1)\mathbb{Z}((s, u)\neq 1)$ , then$\zeta^{*}$(5.5)$\zeta^{*}(S, u)=L_{p}(S,\cdot\omega^{1}-u)$

whereis the $r$ adic -function withcharacter (e.g. cf. [I], p. 18).
$L_{p}(s;\chi)$$L$.
$\mathrm{E}\mathrm{X}\mathrm{A}\mathrm{M}\mathrm{p}\mathrm{L}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{e})k=(1,\frac{p+1\mathrm{E}(}{2})\in x$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{L}\mathrm{e}\mathrm{t}p>3$,$\mathrm{c}\mathrm{h}\dot{\mathrm{a}}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{r}\chi$and$\omega$

be a prime number such that$p\underline{=}3$

$,$

(5.6)

$G_{k}^{*}= \frac{1}{2}h(-p)+\sum t=1\infty\sum_{0<d|t}(\frac{d}{p})q^{\ell}$

As mentioned before, $h(-p)$ is the class number of.

$\mathbb{Q}(\sqrt{-p})$is the Teichm\"uller.

(mod 4). If209
6

Main result.

One of the main purpose of this note is to give a generalization of the abovementioned formula (5.6). It is interesting to us that theformula has asimple form unexpectedly.
As was mentioned earlier, the Fourier expansion of Siegel modular form canbe written as$\mathrm{r}\mathrm{a}\mathrm{e}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$f$$f= \sum_{0\leq T\in\Lambda n}af(T)q\in \mathbb{C}\tau[qij, q_{i^{-1}}j][[q_{1}, ., .

, q_{n}]]$

.

Y-

As an
of the degree one case, one can define the notion of -adic Siegel
modular form as the limit of a sequence of ordinary Siegel modular forms
with rational Fourier coefficients:$\mathrm{a}\mathrm{n}\mathrm{a}1_{0^{\sigma}}\mathrm{y}\circ$

$p$

$f$

$\{f_{m}\}$$q^{-1}ij[\tau_{\in \mathbb{Q}[][}q1, \ldots, q_{n}]]$$f= \sum_{0\leq T\in\Lambda n}a(T)qpqij,$,$f_{m}= \sum_{n}a^{(}0\leq\tau \mathrm{e}\Lambda m)(T)q^{\tau}\in \mathbb{Q}[q_{i}j, q_{i}-1]j[[q_{1}, \ldots, q_{n}]]$,$v_{p}(f-f_{m})$ $:= \inf 0\leq\tau\in\Lambda_{n}v_{p}(a(T)-a(m)(T))arrow+\infty$$(marrow\infty)$.

Our result is as follows:
THBOREM 6.1 Let $p>3$ be aput

$p_{7}\dot{\tau}me$

number such that$k_{m}:=1+ \frac{p-1}{2}\cdot pm-1\in \mathbb{N}$ihen the sequence(6.1)$E_{k}^{*}$$\{k_{m}^{\wedge}\}_{m=1}^{\infty}$$p\equiv 3$(mod 4).

If we,

has the limit $k=(1, L^{\underline{+1}})2\in X$ and$:=$$\lim_{marrow\infty}(\frac{1}{2}\zeta(1-k_{m})E_{k)}(2)m$$=$$\frac{1}{2}h(-p)+$$\sum_{\tau,D(\tau 0\leq\in\Lambda)=-}p\tau \mathrm{O}\iota r0\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(T)\sum_{(0<d|\epsilon\tau)}(\frac{d}{p})q\tau$,where we understand $D(T)=0$ if $\det(\tau)=0$ .
To prove this theorem, we prepare some lemma.

LEMMA 6.2 For non negative integers
and integer
for any$Then_{f}$$p_{7\dot{\mathrm{V}}m}ep>3$$\frac{S_{k_{m}}(p^{h})}{p^{h}}\equiv B_{k_{m}}$whereis theTheorem 6.1.
$B_{k_{m}}$$k_{m^{-}}th$$N_{f}$$h\geq 1_{f}$holds:(6.2)$k,$

we define $S_{k}(N):= \sum_{a=1}^{N}a^{k}$ .
the following congruence relation(modBemoulli number and$p^{h}$),$k_{m}$

is the integer defined in210
PROOF. Let $B_{n}(x)$ be the n-th Bernoulli polynomial. The followving identity iswell known:$S_{k}(N)= \frac{1}{k+1}(Bk+1(N)-B_{k+}1(\mathrm{o}))$(e.g. cf. [I], p.15). Since$B_{k+1}(x)-B_{k}+1(0)=(k+1)\cdot B_{k}\cdot x+\cdot B_{k-1}\cdot x^{2}+\cdots$ ,we have$\frac{S_{k_{m}}(p^{h})}{p^{h}}=B_{k_{m}}+\frac{k_{m}}{2}$, $B_{k_{m}}-1^{\cdot}p+ \frac{k_{m}(k_{m}-1)}{2\cdot 3}h$.

The prime does not appear in the denominator of.
once those of $B_{k_{m}-j}(j\geq 2)$ . This shows (6.2).

$B_{k_{m}-1}$$p$

PROOF of Theorem 6.1.
(6.3)

and appeals at.
$\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\square$Put$E_{k_{m}}$$:= \frac{1}{2}\zeta(1-km)Ek_{m}(2)$We write the Fourier expansion of(6.4)$B_{k_{m}}-2^{\cdot}p^{2}h+\cdots$$E_{k_{m}}$

.

by

.

$E_{k_{m}}= \sum_{0\leq\tau\in\Lambda 2}a((m)\tau)q\in \mathbb{Q}T[q_{12}, q1-1]2[[q1, q2]]$

Moreover, put$\frac{1}{2}h(-p)$

(6.5)

if

$T=O_{2}$,$0< \sum_{d|\epsilon(T)}(\frac{d}{p})$if rank $(T)=1$ ,2

if rank $(T)=2$ and $D(T)=-p$ ,$a(T):=\{$$\sum_{0<d|\epsilon(\tau)}(\frac{d}{p})$$0$otherwise.

Our aim is to show the following:(6.6)$0 \leq T\in\Lambda\inf_{2}v_{p}(a^{(m)}(T)-a(T))arrow+\infty$$(marrow\infty)$.

As a first step, we shall show that
(6.7)$\lim_{marrow\infty}a^{(m)}(o_{2})=\lim_{marrow\infty}(-\frac{B_{k_{m}}}{2k_{m}})=\frac{1}{2}h(-p)$.

Although this is a part of the result (5.6), we shall give a direct proof. By
Kummer’s congruence (3.3),$(1-p^{k_{m}1}-) \frac{B_{k_{m}}}{k_{m}}\equiv(1-p^{k1}-)\iota\frac{B_{k_{l}}}{k_{l}}$(mod$p^{l}$)

211
for $m>l$ (note that $p>3$ ). This means that the sequence $\{(1-p^{k}m-1)Bkm/k_{m\}}$ ,henceconverges in . By Euler’s criterion,$\{B_{k_{m}}/k_{m}\}$$\mathbb{Q}_{p}$$a^{k_{m}}=(a^{R_{\frac{-1}{2}}})^{p^{m-}}1a \equiv(\frac{a}{p})a$(mod$p^{m}$).

Hence we have
(6.8)

$S_{k_{m}}(p^{h})= \sum_{a=1}^{p^{h}}ak_{m}\equiv\sum_{a=1}^{p^{h}}(\frac{a}{p})a=(\mathrm{P}^{-}\sum_{a=1}^{1}(\frac{a}{p})a)ph-1$

for any positive integers $m,$(69)$\mathrm{R}\cdot \mathrm{o}\mathrm{m}$$h$with $m>h,$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{y}\sim$$\frac{S_{k_{m}}(p^{h})}{p^{h}}\equiv\frac{1}{p}(a\sum_{1=}^{p-1}(\frac{a}{p})a)$(mod$p^{m}$(mod$p^{h}$)’(mod $p^{m-h}$ ).

this, we have

(6.10)$\lim_{marrow\infty}\frac{S_{k_{m}}(p^{h})}{p^{h}}=\frac{1}{p}(a\sum_{1=}^{p-1}(\frac{a}{p})a)$for any fixed integer . Using (6.2), we obtain$h$$\lim_{marrow\infty}\frac{B_{k_{m}}}{k_{m}}=\lim Bmarrow\infty k_{m}\equiv\lim_{marrow\infty}\frac{S_{k_{m}}(\mathrm{P}^{h})}{p^{h}}=\frac{1}{p}(p\sum_{a=1}^{1}-(\frac{a}{p})a)$).

This shows(6.11)$\lim_{marrow\infty}\frac{B_{k_{m}}}{k_{m}}=\frac{1}{p}(_{a}\sum_{rightarrow,-1}^{p-}1(\frac{a}{p})a)$.

From the general formula for $h(D)$ ( : fundamental discriminant), we get thefollowing identity:$D$(6.12)$h(-p)=- \frac{1}{p}(_{a=1}^{p-1}\sum x-p(a)a)=-\frac{1}{p}(p\sum_{a=1}^{1}-(\frac{a}{p})a)$(e.g. cf. [Z], \S 9, Satz 3). Combining (6.11) and (6.12), we get (6.7). The second
step is to prove the following: for $T\neq O_{2}$ ,(6.13)$a^{(m)}(T)\equiv a(T)$(mod$p^{m}$).

or equivalently,(6.14)$\mathit{0}_{2}\neq T\in\Lambda\inf_{2}v_{p}(a^{(m)}(T)-a(T))\geq m$.

212First assume that$T$

is rank 1. In this case, by (4.5), we have$a^{(m)}( \tau)=-\frac{B_{k_{m}}}{2k_{m}}\cdot a_{k_{m}}^{(}2)(T)=\sigma_{km^{-1}}(\epsilon(T))$.

Again by Euler’s criterion, we obtain(6.15)(mod$a^{(m)}( \tau)=\sum_{0<d|\epsilon(T)}dkm^{-1}=\sum_{(0<d|_{\mathcal{E}}T)}d\epsilon_{\frac{-1}{2}}pm-1\equiv\sum_{\tau 0<d|\mathcal{E}()}(\frac{d}{p})$Finally we assume thatwritten as$T\in\Lambda_{2}$).

$p^{m}$

is rank 2. By Corollary 4.3, $a^{(m)}(T)$ can be(6.16)$a^{(m)}( \tau)=-\frac{B_{k_{m}}}{2k_{m}}\cdot a_{k_{m}}^{(}(2)T)=\frac{2B_{\dot{k}_{m}-1,\chi\tau}D()}{B_{2k_{m}-2}}\cdot F_{k_{m}}(T)$,.

$F_{k_{m}}(T)=0<d| \sum_{)\zeta(T}dk_{m}-1\sum_{arrow 0<f|(\tau_{)}d}\mu(f)\chi D(\tau)(f)fk_{m}-2\sigma 2km^{-3}(\frac{f(T)}{fd})$

We shall prove the following:(6.17)$\frac{B_{k_{m^{-1}},x_{D(\tau})}}{B_{2k_{m}-2}}\equiv$(mod$p^{m}$).

By definition, the factor of Bernoulli numbers becomes$\frac{B_{k_{m}-\mathrm{i},\chi}D(T)}{B_{2k_{m}-2}}=\frac{B_{2_{\frac{-1}{2}\cdot p^{m-},x_{DT)}}}1(}{B_{()p^{m-1}}p-1}$.

Suppose that $D(T)\neq-p$. By Theorem 3.2, (1), (2) and (3.5), we have
$B_{\epsilon_{\frac{-1}{2}}.m},\equiv 0$(mod$p^{m}$),$pB_{()p}p-1m-1\equiv p-1$

(mod$p^{m}$).

From these formulas, we get$\frac{B_{\mathrm{L}^{-\underline{1}}2}.p^{m}-1\chi D\mathrm{t}T)}{B_{(p-1})\mathrm{p}^{m-}1},\equiv 0$(mod$p^{m}$).

Suppose that $D(T)=-p$. By (3.5) and Theorem 3.2, (3), we have$pB_{L^{-}2}\underline{\iota}_{p,x-_{\mathrm{P}}}.m-1\equiv p-1$

(mod$p^{m}$),$pB_{()}p-1p^{m-}1\equiv p-1$

From these formulas, we obtain$\frac{B_{\epsilon_{\frac{-1}{2}\cdot p\chi-}}m-\iota 1\mathrm{P}}{B_{()p}-1mp-1}\equiv 1$(mod$p^{m}$),(mod$p^{m}$).

213
and this completes the proof of (6.17). Next we shall show
that: if $D(T)=-|)$,
then(6.18)$F_{k_{m}}(T) \equiv 0<d_{1}^{\mathrm{t}}\sum_{\tau\epsilon()}(\frac{d}{p})$(mod$p^{m}$).

In our case, we have $x_{D(T}$ ) $(a)= \chi_{-p}(a)=(\frac{a}{p})$ . Therefore(mod$( \frac{f(T)}{fd})$$F_{k_{m}}(T) \equiv\sum_{0<d|\epsilon(\tau)}(\frac{d}{p})\sum_{f(,p}\mu(f)f^{-}1\sigma_{-}^{*}0<f|\frac{f(T)}{)=^{d}1}1,$

where
that

$\sigma_{-1}^{*}(l)=\sum_{0<d|l},(d,\mathrm{p})=1d-1$(6.19)$\sum$(cf. \S 5). To prove (6.18), it$p^{m}$$\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}_{4}\mathrm{c}\infty$),to$\mathrm{s}\mathrm{l}_{\mathrm{l}\mathrm{t})}\mathrm{w}$$\mu(f)f^{-1}\sigma^{*}-1(\frac{f(T)}{fd})=1$$0<f|\angle\perp T\lrcorner$$(j,p)=^{d}1$

for any

$d$

wvith

$d|\epsilon(T)$. In general, we can prove(6.20)$0_{l}< \sum_{(,p)=}l|m_{1}\mu(l)l^{-}1\sigma-1(\frac{m}{l})=1$

for any $m\in$ N. For any( : prime). Then$m\in \mathbb{N}$$p_{1}^{e_{1\mathrm{e}_{f}}}\cdots p_{r}$$p_{i}$

with $p^{e}||m$ , we put $m_{0}:=m/p^{(arrow}$.

$=$$\neq p$$=$$\sum_{0<l|m}\mu(l)l-1\sigma_{-1}(\frac{m_{0}}{l})$$(l,p)=0<l \sum_{1m_{1}}\mu(l)l^{-}1\sigma_{-1}*(\frac{m}{l})$$=$$\prod_{i=1}^{r}(_{0<l}\sum_{:1p}\mu(l)l-1\sigma-1(\frac{p_{i}}{l}))\vee\cdot--$

The inner sum of the last formula is trivially equal to 1. This shows (6.20).
and (6.18), we obtain$\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\circ\sigma(6.17)$$a^{(m)}(T)\equiv\{$

if$0^{0<}2 \sum_{\epsilon d|(T)}(\frac{d}{p})$$D(T)=-p\}$otherwvise(mod$p^{m}$).

This proves (6.13). If we put $b_{m}:=v_{p}(a^{(m)}(O2)-a(o_{2}))$ , then: by (6.5)(6.7), we have $b_{m}arrow+\infty(marrow\infty)$ . Therefore we obtain$0 \leq T\in\inf_{2}\Lambda v_{p}(a^{(m)}(T)-\alpha(T))\geq\min(m, b_{m})arrow-\dagger\ulcorner\infty$This showa (6.6) and completes the proof of Theorem 6.1.

$(marrow, \infty)$$i1.\mathrm{t}\iota \mathrm{c}1$.
$\square$214
7

Reduction mod$p$of Fourier coefficient of Siegel-Eisenstein series.

By similar argument used in \S 6, we can present an additional formula for the
Fourier coefficient of Siegel-Eisenstein series of degree 2.
The following result is due to Yamaguchi.

THEOREM 7.1 (Yamaguchi [Y]) Let $p>3$ be anumber such that(mod 4). For any $0<T\in\Lambda_{2}$ with $f(T)–1$ , we have$p_{7\dot{\mathrm{V}}}me$$p\equiv$

$3$

(7.1)

(mod )$a_{R\pm\underline{1},2}^{(2)}(T) \equiv-\frac{4pB_{\mathrm{L}_{\frac{-1}{2},\chi}D(T)}}{h(-p)}$(for the definition of

$p$

).

$f(T)_{J}\mathit{8}ee(\mathit{4}\cdot \mathit{6})$REMARK. The right-hand side does not necessarily vanish because there is apossibility that prime appears in the denominator of.
We can genralize the above result.
$p$$B_{E_{\frac{-1}{2},xD()}\tau}$

THEOREM 7.2 Let $p>3$ be a$0<T\in\Lambda_{2}$ , we have(7.2)

$p_{7}\dot{\mathrm{v}}me$$a_{E\pm\underline{1},2}^{(2)}(T) \equiv\frac{4\alpha_{T}}{h(-p)}\sum_{\epsilon 0<d|(\tau)}(\frac{d}{p})$where$p\equiv 3$(mod 4). For any(mod ),

1

if $D(T)=-p$ ,$0$otherwise.

$\alpha_{T}:=\{$PROOF.

number such that$p$

By Corollary 4.3, we can write as$a_{\mathrm{z}\pm}^{()}( \underline{1}T22\cdot 2)=-\frac{2(p+1)\cdot B_{\epsilon}\frac{-1}{2},x_{D}(T)}{B_{L}\underline{+1},2B_{p1}-}\cdot FR\pm\underline{1}(T)$.

Recall(mod ),$B_{\epsilon_{2}} \pm\underline{\iota}\equiv-\frac{h(-p)}{2}\not\equiv 0$$p$(Theorem 3.1, (4)).

This implies(7.3)$a_{\epsilon\pm,2}^{(2)} \underline{1}(T)\equiv\frac{4(p+1)B_{L^{-}2}\underline{\iota}x_{D()}T}{h(-p)\cdot B_{p’-}1}\cdot F_{\frac{\mathrm{p}+1}{2}}(T)$(mod ).
$p$First suppose that $D(T)\neq-p$ . In this case, does not appear in the denominator of(cf. Theorem 3.2, (1)). Then, by the theorem of vonStaudt-Clausen (Theorem 3.1, (2)), the right-hand side of (7.3) is divisible by. Secondly suppose that $D(T)=-p$ . In this case, we have$p$

$B_{\epsilon_{\frac{-\mathrm{I}}{2},\chi}D(T)}$$p$$pB_{p-1}\equiv-1$(mod ),$p.$$pB_{L^{-\underline{\iota}_{\chi-_{\mathrm{P}}}},2},\equiv-1$(mod )$p$

215(cf. (3.5), (3.10)). Therefore, we get(7.4)$\frac{B_{\mathrm{L}_{\frac{-1}{2},x-\mathrm{p}}}}{B_{p-1}}\equiv 1$(mod ).
$p$

So we can rewrite (7.3) as$a_{\epsilon_{\frac{+1)}{2}}}^{(2}(T) \equiv\frac{4\alpha_{T}}{h(-p)}F_{\epsilon\pm\underline{1},2}(T)$(mod ).
$p$

We shall show
(7.5)

$F_{2\pm,2} \underline{1}(T)\equiv\sum_{0<d|\xi(\tau)}(\frac{d}{p})$(mod ).
$p$

The proof of this formula is the same as that of (6.18). In fact, we have
$F_{\epsilon_{2}}\pm\underline{1}(\tau)$$=$$\mu(f)x-p(f)f^{L^{-\underline{3}}}2\sigma p-2(\frac{f(T)}{fd})$$\sum$$\sum_{0<d|\epsilon(\tau)}dR_{\frac{-1}{2}}$$0<f\}\angle\perp_{d}T\Delta$$;\cdot$.

$(\mathrm{m}\mathrm{o}\mathrm{d} p)$$\equiv$$\cdot$.

.

$0<d| \epsilon(\sum_{)T}(\frac{d}{p})\sum_{p)=1}\mu(f)f^{-1*}\sigma-10<f|(f,\angle \mathrm{L}_{d}\tau\lrcorner(\frac{f(T)}{fd})$

We can show by (6.20) that the inner sum is equal to 1. This proves (7.5),
consequently, we get (7.2).

$\mathrm{a}\iota 1(\square 1$

References[BS]

Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press,
New York, 1967.

[Ca]

L. Carlitz,
properties of generalized Bemoulli numbers, J.
Reine Angew. Math. 202 (1959), 174-182.

[Co]

H. Cohen, Sums involving the values at negative integers ofof quadmtic characters, Math. Ann. 217 (1975), 271-285.

[EZ]

M. Eichler and D. B. Zagier, The Theory of Jacobi Forms, Birkh\"auser,
Boston Basel Stuttgart, 1985.

[I].

K. Iwasawa, Lecbures on -adic -functions, Annals of Math. Studies 74,
Princeton Univ. Press, Princeton N. J., 1972.

$A_{7\dot{\eta}}thmeti_{C}$$p$

L-fu.nctions$L$[Kat] H. Katsurada, An explicit formula for Siegel, preprint 1997.

$se7\dot{\eta}es$[Kau] G. Kaufhold, $Di7\dot{7}Chletsche$ Reihe mit Funktionalgleichung in der Theorie
der Modulfunktion 2. Grades, Math. Ann. 137 (1959), 454-476.

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[M]

H. Maass, Die $Fou7\dot{\eta}erkoeffi_{Z}ienten$ der Eisensteinreihen zweiten Grades,
Mat. Fys. Medd. Dan. Vid. Selsk 34 (1964), 1-25.

[Se]

J. -P. Serre, $Fom\iota eS$ modulaires et fonctions z\^eta -adiques, Modular functions of one variable III, 191-268, Lecture Notes in Math. 350,
Verlag, 1973.
$p$$\mathrm{S}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}_{\circ}\sigma \mathrm{e}\mathrm{r}$[Si]

C. L. Siegel, Einf\"uhrung in die Theo
Grades, Math. Ann. 116 (1939), 617-657.

[W]

L. C. Washington, Introduction to Cyclotomic Fields, Springer Verlag,
New York Heidelberg Berlin, 1982.

[Y]

I. Yamaguchi, On Bemoulli numbers and its appl.ication, 1996.

[Z]

D. B.
, Zetafunktionen und quadmtische
Berlin Heidelberg New York, 1981.

$7\dot{\tau}e$$\mathrm{Z}\mathrm{a}_{\mathrm{o}}\sigma \mathrm{i}\mathrm{e}\mathrm{r}$Department of Mathematics
Kinki University
Higashi-Osaka
Osaka 577-8502
Japan-mail: nagaoka@math.kindai.ac.jp$\mathrm{E}$der Modulfunktionen n-ten, Springer Verlag,$K_{\ddot{O}7}per$