Decomposition theorems for Hilbert modular newforms

0.22 MB·English

by  Benjamin Linowitz

DECOMPOSITION THEOREMS FOR HILBERT MODULAR NEWFORMS
BENJAMIN LINOWITZ
1
1
0
2 Abstract. LetSk+( ,Φ)denote the space generatedby Hilbertmodular newforms (over
N
a fixed totally real field K) of weight k, level and Hecke character Φ. We show how to
an decompose Sk+( ,Φ) into direct sums of twisNts of other spaces of newforms. This sheds
N
J light on the behavior of a newform under a character twist: the exact level of the twist of

7 a newform, when such a twist is itself a newform, and when a newform may be realized as
1 the twist of a primitive newform. These results were proven for elliptic modular forms by

Hijikata, Pizer and Shemanske by employing a formula for the trace of the Hecke operator
]
T Tk(n). We obtain our results not by employing a more general formula for the trace of
N Hecke operators on spaces of Hilbert modular forms, but instead by using basic properties
ofnewformswhichwereprovenforellipticmodularformsbyLi,andAtkinandLi,andlater
.
h
extended to Hilbert modular forms by Shemanske and Walling.
t
a
m
[
1. Introduction

1
v
In their paper Twists of Newforms Hijikata, Pizer and Shemanske [4] show how to decom-
4
4 pose spaces of elliptic modular newforms into direct sums of character twists of other spaces
3 of newforms. These decompositions provide important information about the behavior of
3
newforms under character twists; for example, the exact level of the twist of a newform,
.
1
when such a twist is itself a newform, and when a newform may be realized as the twist
0
1 of a primitive newform. The main technique they used to prove these decompositions is
1 Hijikata’s formula [3] for the trace of the Hecke operator T (n) acting on the space of cusp
: k
v forms S (N,φ) of weight k, level N and character φ.
k
i
X
Fix a positive integer N. The Hecke algebra spanned by the T (n) with n coprime to N
k
r
a acting on Sk(N,φ) is semi-simple. Showing that two Hecke-modules A and B are isomorphic
therefore reduces to showing that the trace of T (n) on A equals the trace of T (n) on B
k k
for all n coprime to N. It is in this context that Hijikata’s formula is applied. For instance,
in Theorem 3.2 they take A to be the space S0(N,ωφ) generated by newforms of level N
k
and character ωφ and B to be the space S0(N,ωφ)ω generated by twists (by ω) of newforms
k
of level N and character ωφ. Here ω is a Dirichlet character modulo a power of a prime
dividing N and φ is a Dirichlet character whose conductor is coprime to the conductor of ω.
Hijikata, Pizer and Shemanske use Hijikata’s formula for the trace of T (n) to show that
k
S0(N,ωφ) = S0(N,ωφ)ω.
k ∼ k
Hijikata’s formulas for the trace of Hecke operators apply in much more general contexts
than modular forms on subgroups of SL (Z). For instance, they apply equally well to spaces
2
1
2
of Hilbert modular forms. In theory one could use these more general formulas in order
to extend the results of [4] to the Hilbert modular setting. However the general formulas
are quite complicated, so it is of interest to find a more elementary method of extending
the aforementioned results. In this paper we prove several of the results of [4] for Hilbert
modular forms without appealing to formulas for the traces of Hecke operators. In fact, we
use only the basic properties of newforms which were proven for elliptic modular forms in
the fundamental papers [6] and [1] of Li, and Atkin and Li, and later extended to Hilbert
modular forms by Shemanske and Walling [7]. Thus the results of this paper are new for
Hilbert modular forms over totally real number fields other than Q, and provide simplified
proofs for modular forms over Q (the elliptic modular case).
A sample result is the following (see Section 2 for notation and terminology):
Theorem 1.1. Let be an integral ideal which we decompose as = for a power
0
N N PN P
of a prime ideal p coprime to . Set ν = ord . Let φ be a numerical character modulo
0 p
N P
and Φ be a Hecke character extending φφ which satisfies ν < e(Φ ) = ord (f ) < ν.
N ∞ 2 P p ΦP
Then
S+( ,Φ) = S+(pe(ΦP) ,Ψ2Φ)Ψ,
k N k N0
e(Ψ)=Mν−e(ΦP)
where the sum is taken over all p-primary Hecke characters Ψ with conductor
e(Ψ)=ν−e(ΦP)
pν−e(ΦP) and infinite part Ψ (a) = sgn(a)l for l Zn and a K×.
L ∞ ∈ ∈ ∞
2. Notation and Preliminaries
Forthemostpartwefollowthenotationof[7,8,9]. However, tomakethispapersomewhat
self-contained, we shall briefly review the basic definitions of the functions and operators
which we shall study.
Let K be a totally real number field of degree n over Q with ring of integers , group of
O
units × and totally positive units ×. Fix an embedding a (a(1), ,a(n)) of K into
O O+ 7→ ···
Rn. Let d be the different of K. If q is a finite prime of K, we denote by K the completion
q
of K at q, the valuation ring of K , and π a local uniformizer.
q q q
O
We denote by K the ring of K-adeles and by K× the group of K-ideles. As usual we
A A
view K as a subgroup of K via the diagonal embedding. If α˜ K×, we let α˜ denote the
A ∈ A ∞
archimedean part of α˜ and α˜ the finite part of α˜. If is an integral ideal we let α˜ denote
0 J
J
the -part of α˜.
J
For an integral ideal we define a numerical character φ modulo to be a character
N N
φ : ( / )× C×, and a Hecke character to be a continuous character on the idele class
O N →
group: Φ : K×/K× C×. We denote the induced character on K× by Φ as well. Every
A → A
Hecke character is of the form Φ(α˜) = Φ (α ) where each Φ is a character Φ : K×
ν ν ν ν ν ν −→
C×. The conductor, cond(Φ), of Φ is defined to be the modulus whose finite part is f (see
Φ
Q
[2]) and whose infinite part is the formal product of those archimedean primes ν for which
3
Φ is nontrivial. In the case that f is a power of a single prime q, we define the exponential
ν Φ
conductor e(Φ) to be the integer such that f = qe(Φ). We adopt the convention that φ and
Φ
ψ will always denote numerical characters and Φ and Ψ will denote Hecke characters.
Let GL+(K) denote the group of invertible matrices with totally positive determinant and
2
the complex upper half-plane. Then GL+(K) acts on n via fractional linear transfor-
H 2 H
mations as follows:
a b a(ν)τ +b(ν)
ν
τ , ,
c d 7→ → ··· c(ν)τ +d(ν) ···
(cid:18) (cid:19) (cid:20) (cid:18) ν (cid:19)(cid:21)
Let k = (k ,...,k ) Zn,τ n and set
1 n ∈ + ∈ H
n
(cτ +d)k = (c(ν)τ +d(ν))kν
ν
ν=1
Y
and for A GL+(K)
∈ 2
n
det(A)k = (a(ν)d(ν) b(ν)c(ν))kν.
−
ν=1
Y
For N Z , let Γ denote the kernel of the reduction map SL ( ) SL ( /N ).
+ N 2 2
∈ O → O O
Following Shimura [8, 9], we define M (Γ ) to be the complex vector space of functions
k N
f which are holomorphic on n and at the cusps of Γ such that
N
H
k
f(Aτ) = det(A)−2(cτ +d)kf(τ)
a b
for all A = Γ . Let M = ∞ M (Γ ).
c d ∈ N k N=1 k N
(cid:18) (cid:19)
S
For a fractional ideal and integral ideal we set
I N
−1d-1
Γ ( , ) = A O I : detA × .
0 N I { ∈ d ∈ O+}
(cid:18) N I O (cid:19)
Let θ : × C× be a character of finite order and note that there exists an element
O+ →
m Rn such that θ(a) = aim for all totally positive a. While such an m is not unique, we
∈
shall fix one such m for the remainder of this paper. Let φ be a numerical character modulo
and define M (Γ ( , ),φ,θ) to be the set of f M which satisfy
k 0 k
N N I ∈
k
f(Aτ) = det(A)−2φ(a)θ(detA)(cτ +d)kf(τ)
a b
for all A = Γ ( , ).
c d ∈ 0 N I
(cid:18) (cid:19)
4
Fix a set of strict ideal class representatives ,..., of K, set Γ = Γ ( , ), and put
1 h λ 0 λ
I I N I
h
M ( ,φ,θ) = M (Γ ,φ,θ).
k k λ
N
λ=1
Y
We are interested in studying h-tuples (f ,...,f ) M ( ,φ,θ).
1 h k
∈ N
In order to deal with class number h > 1 we follow Shimura [8, 9] and describe Hilbert
modular forms as functions on an idele group. Let G = GL (K ) and view G = GL (K)
A 2 A K 2
asasubgroupofG viathediagonalembedding. DenotebyG = GL (R)n thearchimedean
A ∞ 2
part of G and by G the subgroup of elements having totally positive determinant. For
A ∞+
an integral ideal of and a prime p, let
N O
a b d-1
Y ( ) = A = Op Op : detA K×,(a , ) = 1 ,
p N { c d ∈ d ∈ p Op N Op }
p p
(cid:18) (cid:19) (cid:18) N O O (cid:19)
W ( ) = x Y ( ) : detx ×
p N { ∈ p N ∈ Op }
and put
Y = Y( ) = G G Y ( ) ,
A ∞+ p
N ∩ × N
!
p
Y
W = W( ) = G W ( ).
∞+ p
N × N
p
Y
Given a numerical character φ modulo define a homomorphism φ : Y C× by setting
Y
N →
a˜
φ ( ∗ ) = φ(a˜ mod ).
Y N
N
(cid:18) ∗ ∗ (cid:19)
˜
Given a fractional ideal of K define = ( ) to be a fixed idele such that = 1 and
ν ν ∞
I I I I
1 0
˜ = . For λ = 1,...,h, set x = G . By the Strong Approximation theorem
IO I λ 0 I˜ ∈ A
λ
(cid:18) (cid:19)
we have
h h
G = G x W = G x−ιW
A K λ K λ
λ=1 λ=1
[ [
where ι denotes the canonical involution on two-by-two matrices.
For an h-tuple (f ,...,f ) M ( ,φ,θ) we define a function f : G C by
1 h k A
∈ N →
f(αx−ιw) = φ (wι)det(w )im(f w )(i)
λ Y ∞ λ | ∞
for α G , w W( ) and i = (i,...,i) (with i = √ 1). Here
K
∈ ∈ N −
a b k aτ +b
fλ | c d (τ) = (ad−bc)2(cτ +d)−kfλ cτ +d .
(cid:18) (cid:19) (cid:18) (cid:19)
5
As in [8, 9], we identify M ( ,φ,θ) with the set of functions f : G C satisfying
k A
N →
(1) f(αxw) = φ (wι)f(x) for all α G ,x G ,w W( ),w = 1
Y K A ∞
∈ ∈ ∈ N
(2) For each λ there exists an element f M such that
λ k
∈
f(x−ιy) = det(y)im(f y)(i)
λ λ |
for all y G .
∞+
∈
Let φ : K× C× be defined by φ (a˜) = sgn(a˜ )k a˜ 2im, where m was defined in the
∞ A → ∞ ∞ | ∞|
definition of θ. We say that a Hecke character Φ extends φφ if Φ(a˜) = φ(a˜ mod )φ (a˜)
∞ N ∞
N
for all a˜ K× ×. If P denotes the K-modulus consisting of the product of all
∈ ∞ × pOp ∞
the infinite primes of K, then any Hecke character Φ extending φφ has conductor dividing
∞
Q
P . Henceforth wewill usethe wordconductor to refer tothefinitepart oftheconductor.
∞
N
If φ is a numerical character modulo where = pa is a power of a prime p and
0
P N P
(p, ) = 1, then by the Chinese Remainder Theorem we have a decomposition φ = φ φ
N0 P N0
where φ is a numerical character modulo and φ is a numerical character modulo . If
P P N0 N0
Φ is a Hecke character extending φ (i.e. trivial infinite part) and Φ is a Hecke character
P P N0
extending φ φ then it is clear that Φ = Φ Φ . Throughout this paper we shall adopt
N0 ∞ P N0
this convention and decompose Hecke characters Φ extending numerical characters modulo
as Φ = Φ Φ where Φ has trivial infinite part.
PN0 P N0 P
Given a Hecke character Φ extending φφ we define an ideal character Φ∗ modulo P
∞ ∞
N
by
Φ∗(p) = Φ(π˜ ) for p ∤ and π˜ = p,
p
N O
Φ∗(a) = 0 if (a, ) = 1
(cid:26) N 6
Observe that for any a˜ K× with (a˜ , ) = 1, Φ(a˜) = Φ∗(a˜ )φ(a˜ )φ (a˜).
∈ A O N O N ∞
For s˜ K×, define fs˜(x) = f(s˜x). The map s˜ f fs˜ defines a unitary representation
∈ A −→ 7→
of K× in M ( ,φ,θ). By Schur’s Lemma the irreducible subrepresentations are all one-
dimeAnsional (ksiNnce K× is abelian). For a character Φ(cid:0) on K(cid:1)×, let M ( ,Φ) denote the sub-
A A k N
space of M ( ,φ,θ) consisting of all f for which fs˜ = Φ(s˜)f and let S ( ,Φ) M ( ,Φ)
k k k
denote the suNbspace of cusp forms. If s K× then fs = f. It followsNthat M⊂ ( ,NΦ) is
k
∈ N
nonempty only when Φ is a Hecke character.
If f = (f ,...,f ) M ( ,φ,θ), then each f has a Fourier expansion
1 h k λ
∈ N
f (τ) = a (0)+ a (ξ)e2πitr(ξτ).
λ λ λ
0≪Xξ∈Iλ
If m is an integral ideal then following Shimura we define the m-th ‘Fourier’ coefficient of
f by
N(m)k0a (ξ)ξ−k−im if m = ξ −1
C(m,f) = 2 λ 2 Iλ ⊂ O
0 otherwise
(cid:26)
where k = max k ,...,k .
0 1 n
{ }
6
Given f M ( ,φ,θ) and y G define a slash operator by setting (f y)(x) = f(xyι).
k A
∈ N ∈ |
For an integral ideal r define the shift operator B by
r
k 1 0
f | Br = N(r)− 20 f | 0 ˜r−1 .
(cid:18) (cid:19)
The shift operator maps M ( ,Φ) to M (r ,Φ) and takes cusp forms to cusp forms.
k k
N N
Further, C(m,f B ) = C(mr−1,f). It is clear that f B B = f B .
| r | r1 | r2 | r1r2
For an integral ideal r the Hecke operator T = TN maps M ( ,Φ) to itself regardless of
r r k N
whether or not (r, ) = 1. This action is given on Fourier coefficients by
N
C(m,f T ) = Φ∗(a)N(a)k0−1C(a−2mr,f).
r
|
m+r⊂a
X
Like the shift operator, T takes cusp forms to cusp forms. Also note that if (a,r) = 1 then
r
B T = T B . Given f S ( ,Φ) we define the annihilator operator A by
a r r a k p
∈ N
f A = f f T B .
p p p
| − | |
Let S−( ,Φ) be the subspace of S ( ,Φ) generated by all g B where g S ( ′,Φ)
k N k N | Q ∈ k N
for some proper divisor ′ of with ′ . This space is invariant under the action of
N N QN | N
the Hecke operators T with (r, ) = 1.
r
N
Shimura defines ((2.28) of [9]) a Petersson inner product f,g for f,g S ( ,Φ). With
k
h i ∈ N
respect to this inner product the Hecke operators satisfy
Φ∗(m) f T ,g = f,g T
m m
h | i h | i
for integral ideals m coprime to . Let S+( ,Φ) denote the orthogonal complement of
S−( ,Φ) in S ( ,Φ). It followNs from oukr dNiscussion above that S+( ,Φ) is invariant
k N k N k N
under the Hecke operators T with (r, ) = 1.
r
N
Definition 2.1. A newform f in S ( ,Φ) is a form in S+( ,Φ) which is a simultaneous
k N k N
eigenform for all Hecke operators T with q a prime not dividing . We say that f is
q
N
normalized if C( ,f) = 1.
O
As in the classical case, if f S ( ,Φ) is a newform with Hecke eigenvalues λ :
k p
∈ N {
pis prime , then C(p,f) = λ C( ,f) for all primes p ∤ .
p
} O N
Since T : q ∤ is commuting family of hermitian operators, S+( ,Φ) has an or-
{ q N} k N
thogonal basis consisting of newforms. If g S−( ,Φ) is a simultaneous eigenform for all
T with q ∤ then there exists a newform∈h kSN+( ′,Φ) with ′ having the same
q N ∈ k N N | N
eigenvalues as g for all such T .
q
Finally, if f,g S ( ,Φ) are both simultaneous eigenforms for all Hecke operators T
k q
∈ N
with q a prime not dividing having the same Hecke eigenvalues, then we say that f is
N
equivalent to g and write f g. If f is a newform and f g, then there exists c C× such
∼ ∼ ∈
that f = cg. This follows from Theorem 3.5 of [7].
7
3. Twists of Newforms
Throughout this section p will denote a fixed prime ideal of .
O
Fix an integral ideal and write = where is the p-primary part of and
0
N N PN P N
( , ) = 1.
0
P N
Fix a space S ( ,Φ) S ( ,φ,m), where Φ is a Hecke character extending φφ .
k k ∞
N ⊂ N
Definition 3.1. If f S ( ,Φ) is a normalized newform and Ψ is a Hecke character then
k
∈ N
we define the twist of f by Ψ, denoted f , by
Ψ
f (x) = τ(Ψ)−1Ψ(detx) Ψ (r)Ψ∗(rf d)f (1 r) (x),
Ψ ∞ Ψ | 0 1 0
r∈f−Ψ1Xd−1/d−1
where τ(Ψ) is the Gauss sum associated to Ψ defined in (9.31) of [8] and the subscript 0
denotes the projection onto the nonarchimedean part.
Proposition 3.2. Let notation be as above and set = lcm ,f f ,f2 . If f S ( ,Φ)
is a normalized newform then f S ( ,Ψ2Φ) and CL(m,f ){=NΨ∗Φ(mΨ)CΨ(m},f) for∈allkinNtegral
Ψ ∈ k L Ψ
ideals m.
(cid:3)
Proof. This is Proposition 4.5 of [9].
The following proposition is trivial to verify using the action of the Hecke operators on
Fourier coefficients.
Proposition 3.3. Let notation be as above and q be a prime with q ∤ f . For f S ( ,Φ)
Ψ k
∈ N
we have f T = Ψ∗(q)(f T ) .
Ψ | q | q Ψ
Although Proposition 3.2 gives an upper bound for the exact level of f , one can obtain
Ψ
better bounds in certain special cases. Of particular interest to us is the case in which
Ψ = Φ . The following proposition gives an improved bound on the level of f in this
P Ψ
special case and generalizes Proposition 3.6 of [1].
Proposition 3.4. Let f be the conductor of Φ . Set
P
if ord (f) < ord ( )
= N p p P
L p if ordp(f) = ordp( )
(cid:26) N P
If f S ( ,Φ) then f S ( ,Φ Φ ).
∈ k N ΦP ∈ k L P N0
Proof. Let α G ,x G and w W( ) with w = 1. We will show that
K A ∞
∈ ∈ ∈ L
f (αxw) = (φ φ ) (wι)f (x).
ΦP N0 P Y ΦP
a˜ ˜d−1˜b
Write w = .
c˜ ˜d˜ d˜
(cid:18) L (cid:19)
8
Let r f−1d−1/d−1 and observe that by the Strong Approximation theorem there exists
∈
an element r′ K such that
∈
(1) ord (r′) 0 for all primes q such that ord (fd) = 0
q q
≥
(2) ord (r′) ord (d) for all primes q such that q d and q = p
q q
≥ − | 6
(3) a r r′(d c d r) d−1
p p p p p p
− − L ∈ O
Note that such an r′ lies in f−1d−1. We claim that such a r′ is uniquely determined in
f−1d−1/d−1. More precisely, suppose that r ,r f−1d−1/d−1 give rise to r0,r1 f−1d−1/d−1.
0 1
∈ ∈
We will show that if r0 +d-1 = r1 +d-1 then r +d-1 = r +d-1. To do this we will suppose
0 1
that (r0 r1) d-1 and show that (r r ) d-1 for all finite primes q. It will then follow
0 1 q
− ∈ − ∈ O
from the local-global correspondence for lattices that (r r ) d-1.
0 1
− ∈
We have two cases to consider.
Case 1 - q = p: Both r and r lie in f−1d-1 and hence in f−1d-1 = ×d-1 d-1 .
6 0 1 Oq Oq Oq ⊂ Oq
It follows that (r r ) d-1 .
0 1 q
− ∈ O
Case 2 - q = p: By condition (3) we have
r a r0(d c d r ) d−1
0 p p p p p 0 p
− − L ∈ O
and
r a r1(d c d r ) d−1 .
1 p p p p p 1 p
− − L ∈ O
Putting these together yields
r (a ) r0(d c d r ) r (a )+r1(d c d r ) d−1 .
0 p p p p p 0 1 p p p p p 1 p
− − L − − L ∈ O
Observe that by definition of and W( ), each of the terms in parentheses lies in ×.
L L Op
We may therefore ease notation by writing u for the parenthesized unit:
i
(3.1) r u r0u r u +r1u d−1 .
0 1 2 1 3 4 p
− − ∈ O
Also observe that
u +d-1 = a +d-1 = u +d-1
1 p p p 3 p
O O O
and
u +d-1 = d +d-1 = u +d-1 .
2 p p p 4 p
O O O
It follows that
r u +d-1 = r a +d-1 ,
0 1 p 0 p p
O O
r u +d-1 = r a +d-1 ,
1 3 p 1 p p
O O
r0u +d-1 = r0d +d-1 ,
2 p p p
O O
and
r1u +d-1 = r1d +d-1 .
4 p p p
O O
9
Suppose that (r0 r1) d-1 d-1 . Then d (r0 r1) d-1 as well. We conclude,
p p p
− ∈ ⊂ O − ∈ O
by Equation 3.1, that (r u r u ) d-1 . This means that a (r r ) d-1 , hence
1 3 0 1 p p 1 0 p
− ∈ O − ∈ O
(r r ) d-1 .
1 0 p
− ∈ O
We have shown that (r0 r1) d-1 implies that (r r ) d-1 for all finite primes q,
0 1 q
− ∈ − ∈ O
hence (r r ) d-1.
0 1
− ∈
We now show that
f (αxw) = (φ φ ) (wι)f (x).
ΦP N0 P Y ΦP
By definition,
(3.2) f (αxw) = τ(Φ )−1Φ (det(αxw)) Φ∗(rfd)f (1 r) (αxw)
ΦP P P P | 0 1 0
r∈f−X1d-1/d-1
(3.3) = τ(Φ )−1Φ (det(x))Φ (det(w)) Φ∗(rfd)f(αxw(1 −r) )
P P P P 0 1 0
r∈f−X1d-1/d-1
Let r′ f−1d-1/d-1 correspond to r (i.e. r′ satisfies the three conditions listed in the first
∈
paragraph of this proof) and w′ be a solution to the matrix equation
1 r 1 r′
(3.4) w − = − w′.
0 1 0 1
(cid:18) (cid:19)0 (cid:18) (cid:19)0
We note that
a˜+c˜ ˜˜dr′ d˜−1˜b+d˜r′ a˜ν c˜ ˜d˜rr′
(3.5) w′ = L − − L
c˜ ˜˜d d˜ c˜ ˜d˜r
(cid:18) L − L (cid:19)
and that the three conditions defining r′ imply that w′ W( ).
∈ N
Substituting equation 3.4 into equation 3.3 yields
(3.6) τ(Φ )−1Φ (det(x))Φ (det(w)) Φ∗(rfd)f(αx 1 −r′ w′)
P P P P 0 1 0
r∈f−X1d-1/d-1 (cid:0) (cid:1)
Because f S ( ,Φ), we may rewrite this as
k
∈ N
(3.7) = τ(Φ )−1Φ (det(x))Φ (det(w)) Φ∗(rfd)φ ((w′)ι)f (1 r′) (x)
P P P P Y | 0 1 0
r∈f−X1d-1/d-1
10
(3.8) = τ(Φ )−1Φ (det(x))Φ (det(w)) Φ∗(rfd)φ (d )φ (d )f (1 r′) (x)
P P P P N0 p P p | 0 1 0
r∈f−X1d-1/d-1
(3.9) = φ (d )τ(Φ )−1Φ (det(x))Φ (det(w)) Φ∗(rfd)φ (d )f (1 r′) (x)
N0 p P P P P P p | 0 1 0
r∈f−X1d-1/d-1
We proceed by rewriting the sum in terms of r′ rather than r. To do this we consider the
expression
Φ∗(rfd)φ (d )
P P p
inside the summation.
As Φ (α˜) = Φ∗(α˜ )φ (α˜) for all α˜ J with (α˜ ,p) = 1 this expression is equal to:
P P OK P ∈ K OK
Φ (r˜fd˜)φ (rf d )φ (d ).
P P p p P p
Setting D = det(w ) = a d , we rewrite this expression as
p p p p
Φ (r˜f˜d)φ (rf d )φ (a D−1) = Φ (r˜f˜d)φ (a rf d )φ (D−1).
P P p p P p p P P p p p P p
Recall the third condition defining r′: a r r′(d c d r) d−1 . This implies
p p p p p p
− − L ∈ O
a rf d r′f d (d c ˜ d r) f ,
p p p p p p p p p p
− − L ∈ O
andinparticular, a rf d d r′f d f . This, alongwiththefactthatΦ (r) = Φ (r′) = 1,
p p p p p p p P P
− ∈ O
shows that we now have
Φ (r′˜fd˜)φ (d r′f d )φ (D−1) = Φ (r′˜fd˜)φ (r′f d )φ (a ) = Φ∗(r′fd)φ (a ).
P P p p p P p P P p p P p P P p
We have shown that Φ∗(rfd)φ (d ) = Φ∗(r′fd)φ (a ).
P P p P P p
We rewrite equation 3.9 as
(3.10) = φ (d )τ(Φ )−1Φ (det(x))Φ (det(w))φ (a ) Φ∗(r′fd)f (1 r′) (x)
N0 p P P P P p P | 0 1 0
r′∈f−X1d-1/d-1
By definition of W( ), det(w) × for all finite primes q. It follows that
L ∈ Oq
Φ (det(w)) = φ (det(w)) = φ (a d ).
P P P p p
We therefore rewrite equation 3.10 as