Note on mod p property of Hermitian modular forms

0.21 MB·

by  Toshiyuki Kikuta

NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR
FORMS
6
1
0 TOSHIYUKIKIKUTAANDSHOYUNAGAOKA
2

p
1
e
S
7 Abstract. Themodpkernelofthethetaoperatoristhesetofmodularforms
2 whoseimageofthetheta operator iscongruent tozeromoduloaprimep. In
the case of Siegel modular forms, the authors found interesting examples of
] suchmodularforms. Forexample,Igusa’soddweightcuspformisanelement
T ofmod23kernelofthetheta operator. Inthispaper,wegivesomeexamples
N whichrepresentelementsinthemodpkernelofthethetaoperatorinthecase
ofHermitianmodularformsofdegree2.
.
h
t
a 1. Introduction
m
Serre [21] developed the theory of p-adic and mod p modular forms and pro-
[
duced several interesting results. In his theory, the Ramanujan operator θ : f =

2 a qn θ(f) := na qn played an important role. The notion of such oper-
n n
v 7−→
ator was extended to the case of Siegel modular forms. The theta operator(gener-
6 P P
alized Ramanujan operator) on Siegel modular forms is defined by
0
5
Θ :F = a(T)qT Θ(F):= det(T) a(T)qT,
3 7−→ ·
0 T T
X X
1. where F = a(T)qT is the Fourier expansion (generalized q-expansion) of F.
0 For a prime number p, the theta operator acts on the algebra of mod p Siegel
6 modular forPms (cf. B¨ocherer-Nagaoka [3]). In our study, we found Siegel modular
1
forms F which satisfy the property
:
v
i Θ(F) 0 (mod p).
X ≡
The space consisting of such Siegel modular forms is called the mod p kernel of
r
a the theta operator. In this terminology, we can say that the Igusa cusp form of
weight35 is anelementofthe the mod23kernelofthe theta operator(cf. Kikuta-
Kodama-Nagaoka [11]). Moreover the theta series attached to the Leech lattice is
also in the mod 23 kernel of the theta operator (cf. Nagaoka-Takemori[19]).
The main purpose of this paper is to extend the notion of the theta operator
to the case of Hermitian modular forms and, to give some examples of Hermitian
modular forms which are in the mod p kernel of the theta operator. The first
half concerns the Eisenstein series. Let Γ2( ) be the Hermitian modular group
K
O
of degree 2 with respect to an imaginary quadratic number field K. Krieg [13]
constructed a weight k Hermitian modular form F which coincides with the
k,K
weight k Eisenstein series E(2) for Γ2( ).
k,K OK
1Newtitle: Onthethetaoperator forHermitianmodularformsofdegree2
1
2 TOSHIYUKIKIKUTAANDSHOYUNAGAOKA
The first result says that the Hermitian modular form F is in the mod p
p+1,K
kernel of the theta operator under some condition on p, namely
Θ(F ) 0 (mod p) (cf. Theorem 3.3).
p+1,K
≡
(2)
Asacorollary,wecanshowthattheweightp+1HermitianEisensteinseriesE
p+1,K
satisfies
Θ(E(2) ) 0 (mod p)
p+1,K ≡
if the class number of K equals one.
In the remaining part, we give various examples which are in the mod p kernel
of the theta operator. The first example we show is related to the theta series
attached to positive definite, even unimodular Hermitian lattice overthe Gaussian
field. Let beapositivedefinite,evenunimodularHermitianlatticeofrankr with
L
theGrammatrixH. WedenotethecorrespondingHermitianthetaseriesofdegree
n by ϑ(n) =ϑ(n). It is known that the rank r is divisible by 4 and ϑ(n) becomes a
L H L
Hermitianmodularformofweightr. Inthecaser=12,wehaveapositivedefinite,
even integral Hermitian lattice of rank 12, which does not have any vector of
C
L
length one. In this paper we call it the Hermitian Leech lattice. The theta series
attached to satisfies
C
L
Θ(ϑ(2)) 0 (mod 11) (cf. Theorem 4.3).
LC ≡
The next example is connected with the Hermitian theta constant. Let be a
E
set of mod 2 even characteristics of degree 2 (cf. 4.3). We consider the theta
§
constant θ (m ). It is known that the function
m
∈E
1
ψ := θ4k
4k 4 m
m∈E
X
defines a Hermitian modular form of weight 4k (cf. Freitag [8]). The final result
can be stated as
Θ(ψ ) 0 (mod 7), Θ(ψ ) 0 (mod 11) (cf. Theorem 4.6).
8 12
≡ ≡
Our proofis basedonthe fact thatthe image of a weightk modular form ofthe
theta operator is congruent to a weight k+p+1 cusp form mod p (cf. Theorem
2.7) and then we use the Sturm bound (Corollary 2.6).
2. Hermitian modular forms
2.1. Notation and definition. The Hermitian upper half-space of degree n is
defined by
1
H := Z Mat (C) (Z tZ)>0 ,
n n
{ ∈ | 2i − }
where tZ is the transposed complex conjugate of Z. The space H contains the
n
Siegel upper-half space of degree n
S :=H Sym (C).
n n∩ n
Let K be an imaginary quadratic number field with discriminant d and ring of
K
integers . The Hermitian modular group
K
O
0 1
Γn(OK):= M ∈Mat2n(OK) | tMJnM =Jn := 1n −0n
(cid:26) (cid:18) (cid:19)(cid:27)
NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS 3
acts on H by fractional transformation
n
A B
H Z M Z :=(AZ+B)(CZ +D)−1, M = Γn( ).
n ∋ 7−→ h i C D ∈ OK
(cid:18) (cid:19)
LetΓ Γn( )be asubgroupoffinite index andν (k Z)anabeliancharac-
K k
⊂ O ∈
ter of Γ satisfying νk νk′=νk+k′. We denote by Mk(Γ,νk) the space of Hermitian
·
modular forms of weight k and character ν with respect to Γ. Namely it consists
k
of holomorphic functions F :H C satisfying
n
−→
F M(Z):=det(CZ +D)−kF(M Z )=ν (M) F(Z)
k k
| h i ·
for all M = ∗∗ Γ. When ν is trivial, we write it by M (Γ) simply. The
CD ∈ k k
subspace S (Γ,ν ) of cusp forms is characterizedby the condition
k k
(cid:0) (cid:1)
tU 0
Φ F 0, (U GL ( )),
|k 0 U ≡ ∈ n OK
(cid:18) (cid:18) (cid:19)(cid:19)
whereΦ is the Siegeloperator. A modularformF M (Γ,ν )is calledsymmetric
k k
∈
if
F(tZ)=F(Z).
We denote by M (Γ,ν )sym the subspace consisting of symmetric modular forms.
k k
Moreover
S (Γ,ν)sym :=M (Γ,ν )sym S (Γ,ν ).
k k k k k
∩
If F M (Γ,ν ) satisfies the condition
k k
∈
F(Z+B)=F(Z) forallB Her ( ),
n K
∈ O
then F has a Fourier expansion of the form
(2.1) F(Z)= a(F;H)exp(2πitr(HZ)),
0≤HX∈Λn(K)
where
Λ (K):= H =(h ) Her (K) h Z, d h .
n jl n jj K jl K
{ ∈ | ∈ ∈O }
We assume that any F Mk(Γ,νk) has the Fourierpexpansion above. For any
∈
subring R C, we write as
⊂
M (Γ,ν ) := F M (Γ,ν ) a(F;H) R ( H Λ (K)) .
k k R k k n
{ ∈ | ∈ ∀ ∈ }
In the Fourier expansion (2.1), we use the abbreviation
qH :=exp(2πitr(HZ)).
The generalized q-expansion F = a(F;H)qH can be considered as an element
ina formalpower seriesring C[[q]] (cf. Munemoto-Nagaoka[16], p.248)from which
P
we have
M (Γ,ν ) R[[q]].
k k R
⊂
Let p be a prime number and Z the local ring at p, namely, ring of p-integral
(p)
rational numbers. For F M (Γ,ν ) (i = 1,2), we write F F (mod p)
i ∈ ki ki Z(p) 1 ≡ 2
when
a(F ;H) a(F ;H) (mod p)
1 2
≡
for all H Λ (K).
n
∈
2.2. Hermitian modular forms of degree 2. In the rest of this paper, we deal
with Hermitian modular forms of degree 2.
4 TOSHIYUKIKIKUTAANDSHOYUNAGAOKA
2.2.1. Eisenstein series. We consider the Hermitian Eisenstein series of degree 2.
E(2) (Z):= detk/2(M)det(CZ +D)−k, Z H ,
k,K ∈ 2
M=X(C∗D∗)
wherek >4isevenandM = ∗∗ runsoverasetofrepresentaivesof ∗∗ Γ2( ).
CD 0∗ \ OK
Then
E(2)(cid:0) M(cid:1) (Γ2( ),det−k/2)sym. (cid:8)(cid:0) (cid:1)(cid:9)
k,K ∈ k OK Q
Moreover, E(2) M (Γ2( ),det−2)sym is constructed by the Maass lift (Krieg
4,K ∈ 4 OK Q
[13]). Foranevenintegerk 4,E(1) =Φ(E(2) )isthenormalizedEisensteinseries
≥ k k,K
Eisenstein series of weight k for SL (Z).
2
2.2.2. StructureofthegradedringinthecaseK =Q(i). Inthissection,weassume
that K =Q(i). In [12], the authors defined some Hermitian cusp forms
χ S (Γ2( ))sym,F S (Γ2( ),det5)sym,F S (Γ2( ))sym,
8 ∈ 8 OK Z 10 ∈ 10 OK Z 12 ∈ 12 OK Z
characterizedby
χ 0, F =6X , F =X ,
8 |S2≡ 10 |S2 10 12 |S2 12
where X (k =10,12) is Igusa’s Siegel cusp form of weight k with integral Fourier
k
coefficients (cf. Kikuta-Nagaoka [12]).
Theorem 2.1. Let K =Q(i). The graded ring
M (Γ2( ),detk/2)sym.
k K
O
k∈Z
M
is generated by
(2) (2)
E , E , χ , F , and F .
4,K 6,K 8 10 12
For the proof, we should consult, for example, Kikuta-Nagaoka[12].
Remark 2.2. E(2) M (Γ2( ))sym, E(2) M (Γ2( ),det3)sym.
4,K ∈ 4 OK Z 6,K ∈ 6 OK Z
2.2.3. Sturm bound. Sturm gave some condition of p-divisibility of the Fourier co-
efficientsofmodularform. Laterthe boundis studiedinthe caseofmodularforms
with severalvariables. For example, the firstauthor [6] studied the bound for Her-
mitianmodular forms of degree2 with respect to K =Q(i) andQ(√3i). However
the statement is incorrect. We correct it here.
We assume that K =Q(i) and use an abbreviation
m a+bi
(2.2) [m,a+bi,n]:= 2 Λ (K).
a−bi n ∈ 2
(cid:18) 2 (cid:19)
We define a lexicographic order for the different element elements
H =[m,a+bi,n], H′ =[m′,a′+b′i,n′]
of Λ (K) by
2
H H′ (1)tr(H)>tr(H′) or
≻ ⇐⇒
(2)tr(H)=tr(H′), m>m′ or
(3)tr(H)=tr(H′), m=m′, a>a′ or
(4)tr(H)=tr(H′), m=m′, a=a′, b>b′.
NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS 5
Let p be a prime number and F M (Γ2( )) . We define the order of F by
∈ k OK Z(p)
ord (F):=min H Λ (K) a(F;H) 0 (mod p) ,
p 2
{ ∈ | 6≡ }
wherethe“minimum”isdefinedinthesenseoftheaboveorder. IfF 0 (mod p),
≡
then we define ord (F)=( ). In the case of K =Q(√3i), we can also define the
p
∞
order in a similar way.
It is easy to check that
Lemma 2.3. The following equality holds.
ord (FG)=ord (F)+ord (G).
p p p
Then we have
Theorem 2.4. Let k be an even integer and p a prime number with p 5. Let
K =Q(i) or K =Q(√3i). For F M (Γ2( ),ν )sym, assume that ≥
∈ k OK k Z(p)
k k k
,2 , if K =Q(i)
8 8 8
ord (F) (cid:20)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:21) ,
p ≻ k ,2 k , k if K =Q(√3i)
9 9 9
(cid:20)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:21)
Then we have ordp(F)=(∞), i.e., F ≡0 (mod p). Here
detk/2 if K =Q(i)
ν =
k (detk if K =Q(√3i)
and [x] inside of the bracket means the greatest integer such that x.
≤
Remark 2.5. For K = Q(i) (resp. K = Q(√3i)) the matrix [k],2[k],[k]
8 8 8
resp. [k],2[k],[k] isthemaximumoftheelementsinΛ (K)oftheform [k], ,[k]
9 9 9 2 (cid:2) 8 ∗ (cid:3)8 ≥
0 (resp. [k], ,[k] 0).
(cid:0) (cid:2) 9 ∗ 9(cid:3)(cid:1)≥ (cid:2) (cid:3)
Proof. of(cid:2)Theorem(cid:3)2.4 Let K = Q(i). We use the induction on the weight k. We
can confirm that it is true for small k. Suppose that the statement is true for any
k with k <k . We shall prove that the statement is true for the weight k .
0 0
Let ord (F) = [m ,α ,n ] [k],2[k],[k] . Applying Theorem 2.1 to F, we
p 0 0 0 ≻ 8 8 8
can write as
(cid:2) (cid:3)
(2.3) F =P(E(2) ,E(2) ,F ,F )+χ G,
4,K 6,K 10 12 8·
where
P Z [x ,x ,x ,x ] and G M (Γ2( ),ν )sym.
∈ (p) 1 2 3 4 ∈ k−8 OK k−8 Z(p)
Now we recall that
(2.4) a(F ;[m,r,n])= a(F;[m,a+bi,n]).
|S2
a+Xbi∈Z[i]
r=2a
4mn−(a2+b2)≥0
Restricting both sides of (2.3) to S , we obtain
2
F =P(G ,G ,6X ,X ),
|S2 4 6 10 12
where G is the Siegel Eisenstein series of weight k and X is Igusa’s cusp form
k k
appeared in 2.2.2. The identity (2.4) implies that
§
k k k
ord (F ) ,2 , .
p |S2 ≻ 8 8 8
(cid:20)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:21)
6 TOSHIYUKIKIKUTAANDSHOYUNAGAOKA
In particular, we have
k k
ord (F ) ,r,
p |S2 ≻ 10 10
(cid:20)(cid:20) (cid:21) (cid:20) (cid:21)(cid:21)
for any r Z. By Theorem 2.4 in Kikuta-Kodama-Nagaoka [11], we have F 0
∈ ≡
(mod p). Therefore P 0 (mod p) as a polynomial and hence
≡
F χ G (mod p).
8
≡ ·
By Lemma 2.3, ord (G) = [m 1,α (1 +i),n 1] because of ord (χ ) =
p 0 0 0 p 8
− − −
[1,1+i,1]. It follows that
ord (G)=[m 1,α (1+i),n 1]
p 0 0 0
− − −
k k k
0 0 0
1,2 (1+i), 1
≻ 8 − 8 − 8 −
(cid:20)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:21)
k 8 k 8 k 8
0 0 0
− ,2 − , − .
≻ 8 8 8
(cid:20)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:21)
By the induction hypothesis, we have G 0 (mod p). This completes the proof in
≡
the case K =Q(i).
TheproofinthecaseofK =Q(√3i)isalmostthesameasthecaseK =Q(i). (cid:3)
Corollary 2.6. Assume that K =Q(i) (resp. Q(√3i)) and p is a prime number
with p 5. If a Hermitian modular form F M (Γ2( ),ν )sym satisfies
≥ ∈ k OK k Z(p)
a(F;H) 0 (mod p)
≡
k k
for all H Λ (K) with tr(H) 2 (resp. tr(H) 2 ), then
2
∈ ≤ 8 ≤ 9
(cid:20) (cid:21) (cid:20) (cid:21)
F 0 (mod p).
≡
2.3. Theta operator. WerecallthattheFourierexpansionofHermitianmodular
form can be regarded as an element of certain formal power series ring C[[q]]. The
theta operator over C[[q]] is defined as
Θ :F = a(F;H)qH Θ(F):= a(F;H) det(H)qH.
7−→ ·
In the case that nX= 1, the theta operator is eXqual to the Ramanujan operator,
which produces several interesting results (cf. Serre [21]). It should be noted that
Θ(F) is not necessarily a Hermitian modular form even if F is.
We fix a prime number p. If F M (Γ,ν ) satisfies
∈ k k Z(p)
Θ(F) 0 (mod p),
≡
then we call F an element of the mod p kernel of the theta operator. Assume that
F = a(F;H)qH M (Γ,ν ) . If there is an integer r(r <n) such that
∈ k k Z(p)
P a(F;H) 0 (mod p)
≡
holds fro all H Λ (K) with rank(H) > r, then F is called a mod p singular
n
∈
Hermitian modular form (e.g. cf. B¨ocherer-Kikuta[2]). It is obvious that, if F is a
mod p singular Hermitian modular form, then F is an element of mod p kernel of
the theta operator.
The main purpose of this paper is to give some examples of Hermitian modular
form in the mod p kernel of the theta operator in the case that n=2.
NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS 7
2.3.1. Basic property of theta operator. As we statedabove,the image Θ(F) isnot
necessarily a Hermitian modular form. However the following result holds:
Theorem 2.7. Assume that K =Q(i) and p is a prime number such that p 5.
≥
For any F M (Γ2( ),detk/2)sym, there is a cusp form
∈ k OK Z(p)
G S (Γ2( ),det(k+p+1)/2)sym
∈ k+p+1 OK Z(p)
such that
Θ(F) G (mod p).
≡
Proof. AcorrespondingstatementinthecaseofSiegelmodularformscanbefound
inB¨ocherer-Nagaoka[3],Theorem4. Theproofherefollowsthesameline. Wecon-
siderthenormailizedRankin-Cohenbracket[F ,F ]forF M (Γ2( ),detki/2)sym
1 2 i ∈ ki OK Z(p)
(i = 1,2) (e.g. cf. Martin-Senadheera [14]). We can show that [F ,F ] becomes a
1 2
Hermitian cusp form
[F ,F ] S (Γ2( ),det(k1+k2+2)/2)sym.
1 2 ∈ k1+k2+2 OK Z(p)
If p is a prime number such that p 5, then there is a Hermitian modular form
G M (Γ2( ),det(p−1)/2)sym≥such that
p−1 ∈ p−1 OK Z(p)
G 1 (mod p) (cf. Kikuta-Nagaoka [12]. Proposition 5).
p−1
≡
For a given F M (Γ2( ),detk/2)sym, we have
∈ k OK Z(p)
Θ(F) [F,G ] (mod p).
p−1
≡
Hence we may put
G:=[F,G ] S (Γ2( ),det(k+p+1)/2)sym.
p−1 ∈ k+p+1 OK Z(p)
(cid:3)
Example 2.8.
Θ(E ) 5E χ +2F (mod 7).
4,K 4,K 8 12
≡ ·
3. Eisenstein case
In this section we deal with the Hermitian modular forms related to the Eisen-
stein series of degree 2.
3.1. Krieg’s result. We denote by h the class number of K and w the order
K K
of the unit group of K.
Given a prime q dividing D := d define the q-factor χ of χ (cf. Miyake
K K q K
−
[15], p.80). Then χ can be decomposed as
K
χ = χ .
K q
qY|DK
We set
a (ℓ):= (1+χ ( ℓ)).
DK q −
qY|DK
Let D =mn with coprime m,n. We set
K
ψ := χ , ψ :=1.
m q 1
q:prime
Y
q|m
8 TOSHIYUKIKIKUTAANDSHOYUNAGAOKA
For H Λ (K) with H =O , we define
2 2
∈ 6
ε(H):=max ℓ N ℓ−1H Λ (K) .
2
{ ∈ | ∈ }
Krieg’s result is stated as follows:
Theorem 3.1. (Krieg [13]) Assume that k 0 (mod w ) and k > 4. Then
K
there exists a modular form F M (Γ (≡ ))sym whose Fourier coefficient
k,K ∈ k 2 OK Q
a(F ;H) is given by
k,K
4k(k 1) D det(H)
B B− dk−1GK k−2; K ·d2 if H >0,
 k· k−1,χK 0<Xd|ε(H) (cid:18) (cid:19)
a(Fk,K;H)=−B2kk dk−1 if rank(H)=1,
0<Xd|ε(H)
1 if H =O ,
where Bm(resp. Bm,χ) is the2 Bernoulli (resp. the generalized Bernoulli) number
and
1
G (s;N):= ψ ( N/d)ψ (d)ds.
K m n
a (N) −
DK 0<Xd|NmnX=DK
(m,n)=1
Remark 3.2. In [13], Krieg stated that the modular form F coincides with the
k,K
weight k Hermitian Eisenstein series (in his notation Ek) for any K. However it
2
is known that it is true only for the case h =1.
K
The first main result is as follows:
Theorem 3.3. Let F be the Hermitian modular form introduced in Theorem
k,K
3.1. If p>3 is a prime number such that χ (p)= 1 and h 0 (mod p), then
K K
− 6≡
Θ(F ) 0 (mod p).
p+1,K
≡
Corollary 3.4. Assume that h = 1 and p > 3 is a prime number such that
K
χ (p)= 1. Then the weight p+1 Hermitian Eisenstein series E(2) satisfies
K − p+1,K
Θ(E(2) ) 0 (mod p).
p+1,K ≡
Remark 3.5. (1) In the above theorem, the weight condition k = p + 1 0
≡
(mod w ) is automatically satisfied. In fact, in the case K = Q(i), the condition
K
χ (p) = 1 implies p 3 (mod 4). Then p+1 0 (mod 4). In the case K =
K
− ≡ ≡
Q(√3i), it follows from the condition χ (p) = 1 that p 1 (mod 3). Since p
K
− ≡ −
is odd, we have p+1 0 (mod 6). Since w = 2 in the other cases, p+1 0
K
≡ ≡
(mod w ) is obvious.
K
(2) It is known that there are infinitely many K and p satisfying
χ (p)= 1 and h 0 (mod p)
K K
− 6≡
(e.g. cf. Horie-Onishi [10]).
(3) Our interest is to construct an element of mod p kernel of the theta operator
with the possible minimum weight (i.e., the weight is its filtration. For the details
on the filtration of the mod p modular forms, see Serre [21]). If we do not restrict
ontheweight, wecanconstructsometrivialexamples inseveralways. Forexample,
the power Fp of a modular form F is such a trivial example. If F is of weight k,
then its weight is pk and this is too large. We suppose that the possible minimum
weight is p+1 for mod p non-singular cases (cf. B¨ocherer-Kikuta-Takemori [1]).
NOTE ON MOD p PROPERTY OF HERMITIAN MODULAR FORMS 9
For the proof of the theorem, it is sufficient to show the following:
Proposition 3.6. Assume that p > 3 is a prime number such that χ (p) = 1
K
−
and h 0 (mod p). Let a(F ;H) be the Fourier coefficient of F at H. If
K k,K k,K
6≡
det(H) 0 (mod p), then
6≡
(3.1) a(F ;H) 0 (mod p).
p+1,K
≡
Proof. By Theorem 3.1, the Fourier coefficient a(F ;H) is expressed as
p+1,K
4(p+1)p D det(H)
a(Fp+1,K;H)= B B dpGK p−1; K ·d2
p+1· p,χK 0<Xd|ε(H) (cid:18) (cid:19)
for H >0. First we look at the factor
4(p+1)p
A:= .
B B
p+1· p,χK
By Kummer’s congruence relation, we obtain
B B 1
p+1 2
= (mod p)
• p+1 ≡ 2 12
B 2h
• pp,χK ≡(1−χK(p))B1,χK =(1−χK(p))−w K (mod p).
K
Since p>3, χ (p)= 1, and h 0 (mod p), the factor A is a p-adic unit.
K K
− 6≡
Next we shall show that, if det(H) 0 (mod p), then the factor
6≡
D det(H)
B := dpGK p−1; K ·d2
0<Xd|ε(H) (cid:18) (cid:19)
satisfies
B 0 (mod p).
≡
We note that
D det(H)
K
χK ·d2 =0 or −1.
(cid:18) (cid:19)
Therefore, it is sufficient to show that, if p∤N, χ (N)=0 or 1, then
K
−
(3.2) G (p 1,N) 0 (mod p).
K
− ≡
To prove the congruence relation (3.2), we need a kind of product formula for
G (s,N). Let S = S be the set of prime number which ramifies in K. For
K K
N N, we decompose N as
∈
N =N N , N = qβq, N = qβq.
1 2 1 2
·
qY∈S qY∈/S
q|N q|N
10 TOSHIYUKIKIKUTAANDSHOYUNAGAOKA
Lemma 3.7. Notation is as above. We have
ψ ( N/d)ψ (d)ds
m n
−
0<Xd|NmnX=DK
(m,n)=1
βq
= χ (q)tqst ψ ( 1)ψ (N ) ψ (q)βqqsβq
K m m 2 n
· − ·
qY|N2Xt=0 mnX=DK qY|m
(m,n)=1 q|N1
ψ (q)βq.
m
·
Yq|n
q|N1
Proof. We have
βq
ψ ( N/d)ψ (d)ds =ψ ( 1) ψ (qβq−t)ψ (qt)qst
m n m m n
− −
0<Xd|N qY|NXt=0
βq
=ψ ( 1) ψ (q)βq χ (q)tqst ψ (q)βqqsβq ψ (q)βq.
m m K n m
− · ·
qY|N2 Xt=0 qY|m Yq|n
q|N1 q|N1
Taking a summation over m (and n), we obtain the desired formula. (cid:3)
We return to the proof of (3.2). From the above lemma, we obtain
G (p 1,N)
K
−
1 βq
= χ (q)tqt(p−1) ψ ( 1)ψ (N )
K m m 2
a (N) · −
DK qY|N2Xt=0 mnX=DK
(m,n)=1
ψ (q)βqq(p−1)βq ψ (q)βq.
n m
· ·
qY|m Yq|n
q|N1 q|N1
Using this formula, we study the p-divisibility of G(p 1,N) separately.
−
(i) The case χ (N)=χ (N )= 1.
K K 2
−
In this case, there is a prime number q N such that q / S, χ (q) = 1, and β
2 K q
| ∈ −
is odd. For this prime number q, we have
βq βq
χ (q)tq(p−1)t ( 1)t =0 (mod p).
K
≡ −
t=0 t=0
X X
This implies that G (p 1,N) 0 (mod p) for this N.
K
− ≡
(ii) The case χ (N)=0.
K
In this case, there is a prime number q S with q N. We may assume that
∈ |
χ (N )= 1. (If χ (N ) = 1, then the proof is reduced to the case (i).) In this
K 2 K 2
−