Congruence between Duke-Imamoglu-Ikeda lifts and non-Duke-Imamoglu-Ikeda lifts

0.21 MB·English

by  Hidenori Katsurada

CONGRUENCE BETWEEN DUKE-IMAMOGLU-IKEDA
LIFTS AND NON-DUKE-IMAMOGLU-IKEDA LIFTS
1
HIDENORI KATSURADA
1
0
2

n
a
J
Abstract. Let k and n be positive even integers. For a Hecke
8
eigenformg intheKohnenplussbspaceofweightk n/2+1/2for
1
Γ0(4), letf be the primitiveformofweight2k n f−orSL2(Z)cor-
−
] responding to g under the Shimura correspondence,and In(g) the
T
Duke-Imamoglu-Ikedaliftofg tothespaceofcuspformsofweight
N k forSpn(Z).Thenwecharacterizeprimeidealsgivingcongruence
h. between In(g) and another cuspidal Hecke eigenform not coming
t from the Duke-Imamoglu-Ikeda lift in terms of the specilal values
a
m of the Hecke L-function and the adjoint L-function of f.
[

1
v
7 1. Introduction
7
3 As is well known, there is a congruence between the Fourier coeffi-
3
cients of Eisenstein series and those of cuspidal Hecke eigenforms for
.
1
SL (Z). This type of congruence is not only interesting in its own right
0 2
1 but also has an importan application to number theory as shown by
1 Ribet [Ri1]. Since then, there have been so many important results
:
v about the congruence of elliptic modular forms (cf. [Hi1],[Hi2],[Hi3].)
i
X In the case of Hilbert modular forms or Siegel modular forms, it is
r sometimes more natual and important to consider the congruence be-
a
tween the Hecke eigenvalues of Hecke eigenforms modulo a prime ideal
P. We call such aP a prime ideal giving thecongruence ora congruence
prime. For a cuspidal Hecke eigenform f for SL (Z), let fˆbe a lift of
2
f to the space M (Γ′) of modular forms of weight l for a modular group
l
Γ′. Here we mean by the lift of f a cuspidal Hecke eigenform whose
ceratin L-function can be expressed in terms of certain L-functions of
f. As typical examples of the lift we can consider the Doi-Naganuma
lift, the Saito-Kurokawa lift, and the Duke-Imamoglu-Ikeda lift. We
then consider the following problem:
Date: January 10, 2011.
2000 Mathematics Subject Classification. Primary 11F67, 11F46, 11F66.
1
2 HIDENORIKATSURADA
Problem. Characterize the prime ideals giving the congruence be-
tween fˆ and a cuspidal Hecke eigenform in M (Γ′) not coming from
l
the lift. In particular characterize them in terms of special values of
certain L-functions of f.
This type of problem was first invetigated in the Doi-Naganuma lift
case by Doi, Hida, and Ishii [D-H-I]. In our previous paper [Ka2], we
considered the relationship between the congruence of cuspidal Hecke
eigenforms with respect to Sp (Z) and the special values of their stan-
n
dard zeta functions. In particular, we proved a conjecture proposed by
Harder [Ha] concerning the congruence between Saito-Kurokawa lifts
and non-Saito-Kurokawa lifts under certain conditions. (See also [Br].)
In this paper, we consider a congruence between Duke-Imamoglu-Ikeda
lifts and non-Duke-Imamoglu-Ikeda lifts, which is a generalization of
our previous conjecture. We explain our main result more precisely.
Let k and n be positive even integers. For a Hecke eigenform g in the
Kohnen plus sbspace of weight k n/2 + 1/2 for Γ (4), let f be the
0
−
primitive form of weight 2k n for SL (Z) corresponding to g under
2
−
the Shimura correspondence, and I (g) the Duke-Imamoglu-Ikeda lift
n
of g to the space of cusp forms of weight k for Sp (Z). Moreover let
n
L(s,f) and L(s,f,Ad) be the Hecke’s L-function and the adjoint L-
function of f, respectively. Then our main result can be roughly stated
as follows:
A prime ideal in the Hecke field of f dividing the numerator of
the algebraic part of L(k,f) n/2−1L(2i + 1,f,Ad) gives a congru-
i=1
ence between I (g) and a Hecke eigenform not coming from the Duke-
n
Q
Imamoglu-Ikeda lift (cf. Theorem 4.7).
The paper is organized as follows. In Section 2, we review the Hecke
theory of Siegel modular forms and the standand L function. In Sec-
tion 3, we review a result concerning the relationship between the con-
gruence of cuspidal Hecke eigenforms with respect to Sp (Z) and the
n
special values of their standardzeta functions. InSection 4, we propose
a conjecture concerning the congruence between Duke-Imamoglu-Ikeda
lifts and non-Duke-Imamoglu-Ikeda lifts, and prove it under a certain
condition. By using this theorem, we can give a non-trivial example
of congruence between the Duke-Imamoglu-Ikeda lift and non-Duke-
Imamoglu-Ikeda lift (cf. Example in Section 4).
The author thanks Professor C. Poor and Professor D. Yuen for
informing him the estimate of the dimension of the space of cusp forms
of weight 18 for Sp (Z). The author also thanks Professor B. Heim,
4
CONGRUENCE 3
Professor H. Hida, Professor S. Yasuda, and Professor T. Yamauchi for
their valuable comments.
Notation. For a commutative ring R, we denote by M (R) the
mn
set of (m,n)-matrices with entries in R. In particular put M (R) =
n
M (R). Here we understand M (R) the set of the empty matrix if
nn mn
m = 0 or n = 0. For an (m,n)-matrix X and an (m,m)-matrix A,
we write A[X] = tXAX, where tX denotes the transpose of X. Let
a be an element of R. Then for an element X of M (R) we often
mn
use the same symbol X to denote the coset X mod aM (R). Put
mn
GL (R) = A M (R) detA R∗ , where detA denotes the
m m
{ ∈ | ∈ }
determinant of a square matrix A, and R∗ denotes the unit group of
R. Let S (R) denote the set of symmetric matrices of degree n with
n
entries in R. Furthermore, for an integral domain R of characteristic
different from 2, let (R) denote the set of half-integral matrices of
n
H
degree n over R, that is, (R) is the set of symmetric matrices of
n
H
degree n whose (i,j)-component belongs to R or 1R according as i = j
2
or not. For a subset S of M (R) we denote by S× the subset of S
n
consisting of non-degenerate matrices. In particular, if S is a subset
of S (R) with R the field of real numbers, we denote by S (resp.
n >0
S ) the subset of S consisting of positive definite (resp. semi-positive
≥0
definite) matrices. Let R′ be a subring of R. Two symmetric matrices
A and A′ with entries in R are called equivalent over R′ with each
other and write A A′ if there is an element X of GL (R′) such that
R′ n
∼
A′ = A[X]. We also write A A′ if there is no fear of confusion. For
∼
X O
square matrices X and Y we write X Y = .
⊥ O Y
(cid:18) (cid:19)
2. Standard zeta functions of Siegel modular forms
For a complex number x put e(x) = exp(2π√ 1x). Furthermore put
−
O 1
J = n − n , where 1 denotes the unit matrix of degree n. For
n 1 O n
n n
(cid:18) (cid:19)
a subring K of R put
GSp (K)+ = M GL (K) J [M] = κ(M)J with someκ(M) > 0 ,
n 2n n n
{ ∈ | }
and
Sp (K) = M GSp (K)+ J [M] = J .
n n n n
{ ∈ | }
Furthermore, put
Γ(n) = Sp (Z) = M GL (Z) J [M] = J .
n 2n n n
{ ∈ | }
4 HIDENORIKATSURADA
A B
We sometimes wrtite an element M of GSp (K) as M =
n C D
(cid:18) (cid:19)
with A,B,C,D M (K). We define a subgroup Γ(n)(N) of Γ(n) as
∈ 2 0
A B
Γ(n)(N) = Γ(n) C O mod N .
0 { C D ∈ | ≡ n }
(cid:18) (cid:19)
A B
LetH beSiegel’supperhalf-space. ForeachelementM =
n C D ∈
(cid:18) (cid:19)
GSp (R)+ and Z H put
n n
∈
M(Z) = (AZ +B)(CZ +D)−1
and
j(M,Z) = det(CZ +D).
Furthermore, for a function f on H and an integer k we define f M
n k
|
as
(f M)(Z) = det(M)k/2j(M,Z)−kf(M(Z)).
k
|
For an integer or half integr l and the subgroup Γ(n)(N) of Γ(n), we
0
denote by M (Γ(n)(N)) the space of (holomorphic) modular forms of
k 0
weight k with respect to Γ(n)(N). We denote by S (Γ(n)(N)) the sub-
0 k 0
space of M (Γ(n)(N)) consisting of cusp forms. Let f be a modular
k 0
(n)
form of weight k with respect to Γ (N). Then f has the following
0
Fourier expansion:
f(Z) = c (A)e(tr(AZ)),
f
A∈HXn(Z)≥0
and in particular if f is a cusp form, f has the following Fourier ex-
pansion:
f(Z) = c (A)e(tr(AZ)),
f
A∈HXn(Z)>0
where tr denotes the trace of a matrix. Let dv denote the invariant
volume element on H defined by
n
dv = det(Im(Z))−n−1 (dx dy ).
1≤j≤l≤n jl jl
∧ ∧
Here for Z H we write Z = (x )+√ 1(y ) with real matrices (x )
n jl jl jl
∈ −
and (y ). For two modular forms f and g of weight l with respect to
jl
(n)
Γ (N) we define the Petersson scalar product f,g by
0 h i
f,g = [Γ(n) : Γ(n)(N)]−1 f(Z)g(Z)det(Im(Z))ldv,
h i 0 ZΓ0(n)(N)\Hn
provided the integral converges.
CONGRUENCE 5
Let L = L (GSp (Q)+,Γ(n)) denote the Hecke algebra over Q
n Q n
associated with the Hecke pair (GSp (Q)+,Γ(n)). Furthermore, let
n
L◦ = L (Sp (Q),Γ(n)) denote the Hecke algebra over Q associated
n Q n
with the Hecke pair (Sp (Q),Γ(n)). For each integer m define an ele-
n
ment T(m) of L by
n
T(m) = Γ(n)(d ... d e ... e )Γ(n),
1 n 1 n
⊥ ⊥ ⊥ ⊥ ⊥
d1,...,dXn,e1,...,en
where d ,...,d ,e ,...,e run over all positive integer satisfying
1 n 1 n
d d , e e (i = 1,...,n 1),d e ,d e = m (i = 1,....,n).
i i+1 i+1 i n n i i
| | − |
Furthermore, for i = 1,...,n and a prime number p put
T (p2) = Γ(n)(1 p1 p21 p1 )Γ(n),
i n−i i n−i i
⊥ ⊥ ⊥
and (p−1) = Γ(n)(p−11 )Γ(n). As is well known, L is generated over
n n
Q by all T(p),T (p2) (i = 1,...,n), and (p−1). We denote by L′ the
i n
subalgebra of L generated by over Z by all T(p) and T (p2) (i =
n i
1,...,n). Let T = Γ(n)MΓ(n) be an element of L C. Write T as
n
⊗
T = Γ(n)γ and for f M (Γ(n)) define the Hecke operator T
γ k k
∪ ∈ |
associated to T as
f T = det(M)k/2−(n+1)/2 f γ.
k k
| |
γ
X
We call this action the Hecke operator as usual (cf. [A].) If f is an
eigenfunction of a Hecke operator T L C, we denote by λ (T) its
n f
∈ ⊗
eigenvalue. Let L be a subalgebra of L . We call f M (Γ(n)) a Hecke
n k
∈
eigenform for L if it is a common eigenfunction of all Hecke operators
in L. In particular if L = L we simply call f a Hecke eigenform. Fur-
n
thermore, we denote by Q(f) the field generated over Q by eigenvalues
of all T L as in Section 1. As is well known, Q(f) is a totally real
n
∈
algebraic number field of finite degree. Now, first we consider the inte-
grality of the eigenvalues of Hecke operators. For an algebraic number
field K, let O denote the ring of integers in K. The following assertion
K
has been proved in [Mi2] (see also [Ka2].)
Theorem 2.1 Let k n+1. Let f S (Γ(n)) be a common eigen-
k
≥ ∈
form in L′ . Then λ (T) belongs to O for any T L′ .
n f Q(f) ∈ n
Let L = L(GSp (Q)+ GL (Z[p−1]),Γ(n)) be the Hecke alge-
np n 2n
∩
bra associated with the pair (GSp (Q)+ GL (Z[p−1]),Γ(n)). L
n 2n np
∩
can be considered as a subalgebra of L , and is generated over Q by
n
T(p) and T (p2) (i = 1,2,..,n), and (p−1). We now review the Sa-
i
take p-parameters of L ; let P = Q[X±,X±,...,X±] be the ring of
np n 0 1 n
6 HIDENORIKATSURADA
Laurent polynomials in X ,X ,...,X over Q. Let W be the group
0 1 n n
of Q-automorphisms of P generated by all permutations in variables
n
X ,....,X and by the automorphisms τ ,....,τ defined by
1 n 1 n
τ (X ) = X X ,τ (X ) = X−1,τ (X ) = X (j = i).
i 0 0 i i i i i j j 6
Furthermore, a group W˜ isomorphic to W acts on the set T =
n n n
(C×)n+1 in a way similarly to above. Then there exists a Q-algebra
isomorphism Φ , called the Satake isomorphism, from L to the W -
np np n
invariant subring PWn of P . Then for a Q-algebra homomorphism λ
n n
from L to C, there exists an element (α (p,λ),α (p,λ),...,α (p,λ))
np 0 1 n
of T satisfying
n
λ(Φ−1(F(X ,X ,...,X ))) = F(α (p,λ),α (p,λ),...,α (p,λ))
np 0 1 n 0 1 n
for F PWn. The equivalence class of (α (p,λ),α (p,λ),...,α (p,λ))
∈ n 0 1 n
under the action of W˜ is uniquely determined by λ. We call this the
n
Satake parameters of L determined by λ.
np
Now assume that an element f of M (Sp (Z)) is a Hecke eigen-
k n
form. Then for each prime number p, f defines a Q-algebra homo-
morphism λ from L to C in a usual way, and we denonte by
f,p np
α (p),α (p),....,α (p) the Satake parameters of L determined by f.
0 1 n np
We then define the standard zeta function L(s,f,St) by
n
L(s,f,St) = (1 p−s)(1 α (p)p−s)(1 α (p)−1p−s) −1.
i i
{ − − − }
p i=1
YY
Let f(z) = a(A)e(tr(Az)) be a Hecke eigenform in S (Γ(n)).
k
A∈Hn(Z)>0
For a positive iPnteger m k n such that m n mod 2 put
≤ − ≡
Λ(m,f,St) = ( 1)n(m+1)/2+12−4kn+3n2+n+(n−1)m+2
−
n
L(m,f,St)
Γ(m+1) Γ(2k n i) .
× − − < f,f > π−n(n+1)/2+nk+(n+1)m
i=1
Y
Thenthefollowing theoremisduetoB¨ocherer [B2]andMizumoto [Mi].
Theorem 2.2. Let l,k and n be a positive integers such that ρ(n)
≤
l k n, where ρ(n) = 3, or 1 according as n 1 mod 4 and n 5,
≤ − ≡ ≥
or not. Let f S (Γ(n)) be a Hecke eigenform. Assume that all the
k
∈
Fourier coefficients of f belong to Q(f). Then Λ(m,f,St) belongs to
Q(f).
For later purpose, we consider a special element in L ; the polyno-
np
mial X2X X X n (X +X−1) is an element of PWn, and thus
0 1 2··· n i=1 i i n
P
CONGRUENCE 7
we can define an element Φ−1(X2X X X n (X +X−1)) of L ,
np 0 1 2··· n i=1 i i np
which is denoted by r .
1
P
Proposition 2.3. Under the above notation the element r belongs
1
to L′ , and we have
n
n
λ (r ) = pn(k−(n+1)/2) (α (p)+α (p)−1).
f 1 i i
i=1
X
Proof. By a careful analysis of the computation in page 159-160 of
[A], we see that r is a Z-linear combination of T (p2) (i = 1,...,n), and
1 i
therefore we can prove the first assertion. Furthermore, by Lemma
3.3.34 of [A], we can prove the second assertion.
3. Congruence of modular forms and special values of
the standard zeta functions
In this section we review a result concerning the congruence between
the Hecke eigenvalues of modular forms of the same weight following
[Ka2]. Let K be an algebraic number field, and O = O the ring
K
of integers in K. For a prime ideal P of O, we denote by O the
(P)
localization of O at P in K. Let A be a fractional ideal in K. If A = PeB
with BO = O we write ord = e. We simply write ord (c) =
(P) (P) P P
ord ((c)) for c K. Now let f be a Hecke eigenform in S (Γ(n)) and
P k
∈
M be a subspace of S (Γ(n)) stable under Hecke operators T L .
k n
∈
Assume that M is contained in (Cf)⊥, where (Cf)⊥ is the orthogonal
complement of Cf in S (Γ(n)) with respect to the Petersson product.
k
Let K be an algebraic number field containing Q(f). A prime ideal
P of O is called a congruence prime of f with respect to M if there
K
exists a Hecke eigenform g M such that
∈
λ (T) λ (T) mod P˜
f g
≡
for any T L′ , where P˜ is the prime ideal of O lying above P. If
∈ n KQ(g)
M = (Cf)⊥, we simply call P a congruence prime of f.
Now we consider the relation between the congruence primes and
the standard zeta values. To consider this, we have to normalize
the standard zeta value Λ(l,f,St) for a Hecke eigenform f because
it is not uniquely determined by the system of Hecke eigenvalues of f.
We note that there is no reasonable normalization of cuspidal Hecke
eigenform in the higher degree case unlike the elliptic modular case.
Thus we define the following quantities: for a Hecke eigenform f(z) =
8 HIDENORIKATSURADA
c (A)e(tr(Az)) in S (Γ(n)), let I be the O -module generated
A f k f Q(f)
by all c (A)′s. Assume that the following:
f
P
(*) There exists a complex number c such that all the Fourier coeffi-
cients cf belongs to Q(f).
Then I is a fractional ideal in Q(f),and therefore, so is Λ(l,f,St)I2
f f
if l satisfies the condition in Theorem 2.2. We note that this fractional
ideal does not depend on the choice of c. We also note that the value
N (Λ(l,f,St))N(I )2 does not depend on the choice of c, where
Q(f) f
N(I ) is the norm of the ideal I . Then, by [Kat2], we have
f f
Theorem 3.1. Let f be a Hecke eigenform in S (Γ(n)). Assume the
k
condition (*). Let l be a positive integer satisfying the condition in The-
orem2.2. LetP be aprime idealofO.Assume thatord (Λ(l,f,St)I2) <
P f
0 and that it does not divide (2l 1)!. Then P is a congruence prime
−
of f. In particular, if a rational prime number p divides the denomina-
tor of N (Λ(l,f,St))N(I )2, then p is divisible by some congruence
Q(f) f
prime of f.
Remark. If the multiplicity one property holds for S (Γ(n)), the
k
condition (*) holds for any cuspidal Hecke eigenform in S (Γ(n)). In
k
[Kat3], we have treated the case where the condition (*) does not nec-
essarily hold.
Now for a Hecke eigenform f in S (Γ(n)), let T denote the subspace
k f
of S (Γ(n)) spanned by all Hecke eigenforms with the same system of
k
the Hecke eigenvalues as f.
Corollary. In addition to the above notation and the assumption,
assume that S (Γ(n)) has the multiplicity one property. Then P is a
k
congruence prime of f with respect to T⊥. In particular, if a rational
f
prime number p divides the denominator of N (Λ(l,f,St))N(I )2,
Q(f) f
then p is divisible by some congruence prime of f with respect to T⊥
f
4. Congruence between Duke-Imamoglu-Ikeda lifts and
non-Duke-Imamoglu-Ikeda lifts
In this section, we consider the conguence between Duke-Imamoglu-
Ikeda lifts and non-Duke-Imamoglu-Ikeda lifts. Throughout this sec-
tion and the next, we assume that n and k are even positive inte-
gers. Let g be a Hecke eigenform belonging to the Kohnen plus space
S+ (Γ (4)). Then g has the following Fourier expansion:
k−n/2+1/2 0
g(z) = c (e)e(ez),
g
e
X
CONGRUENCE 9
whereerunsoverallpositiveintegerssuchthat( 1)k−n/2e 0,1mod4.
− ≡
Moreover let
∞
f(z) = c (m)e(mz)
f
m=1
X
be the primitive form in S (SL (Z)) corresponding to g via the
2k−n 2
Shimura correspondence. For a Dirichlet character χ, we then define
the L-function L(s,f,χ) of f twisted by χ by
L(s,f,χ) = (1 χ(p)β pk−n/2−1/2−s)(1 χ(p)β−1pk−n/2−1/2−s) −1,
{ − p − p }
p
Y
whereβ isanon-zerocomplexnumbersuchthatβ +β−1 = p−k+n/2+1/2c (p).
p p p f
We simply write L(s,f) as L(s,f,χ) if χ is the principal character. We
also define the adjoint L-function L(s,f,Ad) of f by
L(s,f,Ad) = (1 β2p−s)(1 β−2p−s)(1 p−s) −1.
{ − p − p − }
p
Y
We note that L(s,f,Ad) coincides with L(s,f,St). Now we put
c (T) = c ( d ) (pk−n/2−1/2β )νp(fT)F (T,p−(n+1)/2β−1).
In(g) g | T| p p p
p
Y
We note that c (T) does not depend on the choice of β . Define a
In(g) p
Fourier series I (g)(Z) by
n
I (g)(Z) = c (T)e(tr(TZ)).
n In(g)
T∈HXn(Z)>0
In [Ik1] Ikeda showed that I (g)(Z) is a cusp form of weight k with
n
respect to Γ(n) and a Hecke eigenform for L◦ such that
n
n
L(s,I (g),St) = ζ(s) L(s+k i,f).
n
−
i=1
Y
This was first conjecture by Duke and Imamoglu. Thus we call I (g)
n
the Duke-Imamoglu-Ikeda lift of g (or of f). We note that we have
Q(g) = Q(I (g)) = Q(f). Furthermore, we have I = I , where I
n g In(g) g
is the O -module generated by all the Fourier coefficients of g.
Q(f)
Now to consider a congurence between Duke-Imamoglu-Ikeda lifts
and non-Duke-Imamoglu-Ikeda lifts, first we prove the following:
Proposition 4.1 I (g) is a Hecke eigenform.
n
We note that Ikeda proved in [Ik1] that I (g) is a Hecke eigenform
n
for L◦ but has not proved that it is a Hecke eigenform for L .∗ We
n n
∗This was pointed to us by B. Heim (see [He].)
10 HIDENORIKATSURADA
also note that an explicit form of the spinor L-function of I (g) was
n
obtained by Murakawa [Mu] and Schmidt [Sch] assuming that I (g) is
n
a Hecke eigenform.
Proof of Proposition 4.1. We have only to prove that I (g) is
n
an eigenfunction of T(p) for any prime p. The proof may be more or
less well konwn, but for the covenience of the readers we here give the
proof. For a modular form
F(Z) = c (B)e(tr(BZ)),
F
B
X
(p)
let c (B) be the B-th Fourier coefficient of F T(p). Then for any
F |
positive definite matrix B we have
c(p)(B) = pnk−n(n+1)/2 dndn−1 d
F 1 2 ··· n
d1|dX2|···|dn|p
detD−kc (p−1A[tD]),
F
×
D∈Λn(dX1⊥···dn)Λn
where Λ = GL (Z).
n n
Now let E (Z) be the Siegel Eisenstein series of degree n and of
n,k
weight k defined by
E (Z) = j(γ,Z)−k.
n,k
γ∈Γ∞X(n)\Γ(n)
For k n + 1, the Siegel Eisenstein series E (Z) is a holomorphic
n,k
≥
modular form of weight k with respect to Γ(n). Furthermore, E (Z)
n,k
is a Hecke eigenform and in particular we have
E T(p)(Z) = h (pk)E (Z),
n,k n,p n,k
|
where
n
hn,p(X) = 1+ p−Prj=1ijXr.
Xr=1 1≤i1<X···<ir≤n
Let c (B) be the B-th Fourier coefficient of E (Z). Then we have
n,k n,k
h (pk)c (B) = pnk−n(n+1)/2 dndn−1 d
n,p n,k 1 2 ··· n
d1|dX2|···|dn|p
detD−kc (p−1B[tD]).
n,k
×
D∈Λn(dX1⊥···dn)Λn
Let B be positive definite. Then we have
c (B) = a (det2B)k−(n+1)/2L(k n/2,χ ) F (B,p−k),
n,k n,k B q
−
q
Y