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# Zeta functions and Eisenstein series on classical groups

## Abstract

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

## Section 1.

Given a reductive algebraic group *G* over
an algebraic number field, we denote by *G*_{A},
*G*_{a}, and *G*_{h} its
adelization, the archimedean factor of *G*_{A}, and
the nonarchimedean factor of *G*_{A}. We take an
open subgroup *D* of *G*_{A} of the form
*D* = *D*_{0}*G*_{a}
with a compact subgroup *D*_{0} such that
*D*_{0} ∩ *G*_{a} is maximal
compact in *G*_{a}. Choosing a specific type of
representation of *D*_{0} ∩
*G*_{a}, we can define automorphic forms on
*G*_{A} as usual. For simplicity we consider here the
forms invariant under *D*_{0} ∩
*G*_{h}. Each Hecke operator is given by
*D*τ*D*, with τ in a subset 𝔛 of
*G*_{A}, which is a semigroup containing *D*
and the localizations of *G* for almost all
nonarchimedean primes. Taking an automorphic form **f**
such that **f**|*D*τ*D* =
λ(τ)**f** with a complex number λ(τ) for every
τ ∈ 𝔛 and a Hecke ideal character χ of *F*, we put
1.1
where ν_{0}(τ) is the denominator ideal of τ and
*N*(ν_{0}(τ)) is its norm. Now our first main
result is that if *G* is symplectic, orthogonal, or unitary,
then
1.2
where Λ(*s*, χ) is an explicitly determined product
of *L*-functions depending on χ,
*W*_{p} is a polynomial determined for
each *v* ∈ **h** whose constant term is 1, and
p runs over all the prime ideals of the basic number
field. This is a purely algebraic result concerning only nonarchimedean
primes.

Let *Z*(*s, ***f**, χ) denote the right-hand side
of Eq. 1.2. As our second main result, we obtain a product
𝔊(*s*) of gamma factors such that 𝔊*Z* can be
continued to the whole *s*-plane as a meromorphic function
with finitely many poles, when *G* is a unitary group of an
arbitrary signature distribution over a CM field, and **f**
corresponds to holomorphic forms.

Now these problems are closely connected with the theory of Eisenstein
series *E* on a group *G*′ in which *G* is
embedded. To describe the series, let ℨ′ denote the symmetric space
on which *G*′ acts. Then the series as a function of (*z,
s*) ∈ ℨ′ × **C** can be given (in the classical style)
in the form
1.3
where Γ is a congruence subgroup of *G*′, and
*P* is a parabolic subgroup of *G*′ which is a
semidirect product of a unipotent group and *G* ×
*GL*_{m} with some *m*. The adelized version
of δ will be explicitly described in Section 5. Now our third main
result is that there exists an explicit product 𝔊′ of gamma factors
and an explicit product Λ′ of *L*-functions such that
𝔊′(*s*)Λ′(*s*)*Z*(*s, ***f**,
χ)*E*(*z, s*; **f**, χ) can be continued
to the whole *s*-plane as a meromorphic function with finitely
many poles.

Though the above results concern holomorphic forms, our method is applicable to the unitary group of a totally definite hermitian form over a CM field. In this case, we can give an explicit class number formula for such a hermitian form, which is the fourth main result of this paper.

## Section 2.

For an associative ring *R* with identity
element, we denote by *R*^{×} the group of all its
invertible elements and by
*R*_{n}^{m} the
*R*-module of all *m* × *n* matrices
with entries in *R*. To indicate that a union
*X* = ∪_{i∈I}
*Y*_{i} is disjoint, we write *X* =
⊔_{i∈I} *Y*_{i}.

Let *K* be an associative ring with identity element and an
involution ρ. For a matrix *x* with entries in *K*,
we put *x** = ^{t}*x*^{ρ}, and
*x̂* = (*x**)^{−1} if
*x* is square and invertible. Given a finitely generated left
*K*-module *V*, we denote by
*GL*(*V*) the group of all *K*-linear
automorphisms of *V*. We let *GL*(*V*) act
on *V* on the right; namely we denote by *w*α the
image of *w* ∈ *V* under α ∈
*GL*(*V*). Given ɛ = ±1, by an ɛ-hermitian form
on *V*, we understand a biadditive map ϕ:*V*
× *V* → *K* such that ϕ(*x,
y*)^{ρ} = ɛϕ(*y, x*) and ϕ(*ax,
by*) = *a*ϕ(*x, y*)*b*^{ρ}
for every *a, b* ∈ *K*. Assuming that ϕ is
nondegenerate, we put
2.1
Given (*V*, ϕ) and (*W*, ψ), we can define
an ɛ-hermitian form ϕ ⊕ ψ on *V* ⊕ *W* by
2.2
We then write (*V* ⊕ *W*, ϕ ⊕ ψ) =
(*V*, ϕ) ⊕ (*W*, ψ). If both ϕ and ψ are
nondegenerate, we can view *G*^{ϕ} ×
*G*^{ψ} as a subgroup of
*G*^{ϕ⊕ψ}. The element (α, β) of
*G*^{ϕ} × *G*^{ψ} viewed as an
element of *G*^{ϕ⊕ψ} will be denoted by α ×
β or by (α, β). Given a positive integer *r*, we put
*H*_{r} =
*I*′_{r} ⊕
*I*_{r}, *I*_{r} =
*I*′_{r} =
*K*_{r}^{1} and
2.3
We shall always use *H*_{r},
*I*′_{r},
*I*_{r}, and η_{r} in this
sense. We understand that *H*_{0} = {0} and
η_{0} = 0.

Hereafter we fix *V* and a nondegenerate ϕ on *V*,
assuming that *K* is a division ring whose characteristic is
different from 2. Let *J* be a *K*-submodule of
*V* which is totally ϕ-isotropic, by which we mean that
ϕ(*J, J*) = 0. Then we can find a decomposition
(*V*, ϕ) = (*Z*, ζ) ⊕ (*H*, η) and
an isomorphism *f* of (*H*, η) onto
(*H*_{r}, η_{r}) such that
*f*(*J*) = *I*_{r}. In this
setting, we define the parabolic subgroup
of *G*^{ϕ}
relative to *J* by
2.4
and define homomorphisms
→
*G*^{ζ} and
→
*GL*(*J*) such that *z*α −
*z*π_{ζ}^{ϕ} (α) ∈ *J* and
*w*α =
*w*λ_{J}^{ϕ}(α)
if *z* ∈ *Z*, *w* ∈
*J*, and α ∈

Taking a fixed nonnegative integer *m*, we put
2.5
We can naturally view *G*^{ψ} ×
*G*^{ϕ} as a subgroup of
*G*^{ω}. Since *W* = *V* ⊕
*H*_{m}, we can put *X* = *V*
⊕ *H*_{m} ⊕ *V* with the first summand
*V* in *W*, and write every element of *X*
in the form (*u, h, v*) with (*u, h*) ∈
*V* ⊕ *H*_{m} = *W* and
*v* ∈ *V*. Put
Observing that *U* is totally ω-isotropic, we can
define

Proposition 1. *Let* λ(ϕ) *be the maximum
dimension of totally* ϕ-*isotropic K-submodules of V.
Then*2.6*has exactly* λ(ϕ) *orbits*.
*Moreover*,
2.7*with* ξ *running over G*^{ϕ}*and* β *over**G*^{ψ}, *where
**H* = *H*_{m}*and I = I _{m}.*

In fact, we can give an explicit set of representatives
{τ_{e}}_{e=1}^{λ(ϕ)}
for Eq. 2.6 and also an explicit set of representatives for
*P*_{U}^{ω}/*P*_{U}^{ω}τ_{e}[*G*^{ψ}
× *G*^{ϕ}] in the same manner as in Eq. 2.7.
This proposition plays an essential role in the analysis of our
Eisenstein series *E*(*z, s*; **f**, χ).

## Section 3.

In this section, *K* is a locally compact
field of characteristic 0 with respect to a discrete valuation. Our aim
is to establish the Euler factor *W*_{p}
of Eq. 1.2. We denote by r and q
the valuation ring and its maximal ideal; we put *q* =
[r:q] and |*x*| =
*q*^{−ν} if *x* ∈ *K* and
*x* ∈ π^{ν} r^{×}
with ν ∈ **Z**. We assume that *K* has an
automorphism ρ such that ρ^{2} = 1, and put
*F* = {*x* ∈ *K* |
*x*^{ρ} = *x*}, g =
*F* ∩ r, and d^{−1} =
{*x* ∈ *K* |
Tr_{K/F} (*x*r)
⊂ g} if *K* ≠ *F*. We
consider (*V*, ϕ) as in Section 2 with *V* =
*K*_{n}^{1} and ϕ defined by
ϕ(*x, y*) = *x*ϕ*y** for *x,
y* ∈ *V* with a matrix ϕ of the form
3.1
where *t* = *n* − 2*r*. We
assume that θ is anisotropic and also that
3.2a
3.2b
Thus our group *G*^{ϕ} is orthogonal,
symplectic, or unitary. The element δ of Eq. 3.2b can be
obtained by putting δ = *u* −
*u*^{ρ} with *u* such that r
= g[*u*]. We include the case
*rt* = 0 in our discussion. If *t* = 0, we
simply ignore θ; this is always so if *K* =
*F* and ɛ = −1. We have ϕ = θ if *r* = 0.

Denoting by {*e*_{i}} the standard basis of
*K*_{n}^{1}, we put
Then *G*^{ϕ} =
*P*_{J}^{ϕ}*C*.
We choose
{*e*_{r+i}}_{i=1}^{t}
so that *N* =
∑_{i=1}^{t}r*e*_{r+i}.
Then we can find an element λ of
r_{t}^{t} such that
3.3
Put
3.4
We can write every element of
*P*_{J}^{ϕ} in the form
3.5
If *t* = 0, we simply ignore *b, e*, and
*f*, so that ξ =
[ ];
we have ξ = *e* if *r* = 0.

We consider the Hecke algebra ℜ(*E,
GL*_{r}(*K*)) consisting of all formal finite
sums ∑*c _{x}ExE* with

*c*

_{x}∈

**Q**and

*x*∈

*GL*

_{r}(

*K*), with the law of multiplication defined as in ref. 1. Taking

*r*indeterminates

*t*

_{1}, … ,

*t*

_{r}, we define a

**Q**-linear map 3.6 as follows; given

*ExE*with

*x*∈

*GL*

_{r}(

*K*), we can put

*ExE*= ⊔

_{y}

*Ey*with upper triangular

*y*whose diagonal entries are π

^{e1}, … , π

^{er}with

*e*

_{i}∈

**Z**. Then we put 3.7 Next we consider the Hecke algebra ℜ(

*C, G*

^{ϕ}) consisting of all formal finite sums ∑

*c*τ

_{τ}C*C*with

*c*

_{τ}∈

**Q**and τ ∈

*G*

^{ϕ}. We then define a

**Q**-linear map 3.8 as follows; given

*C*τ

*C*with τ ∈

*G*

^{ϕ}, we can put

*C*τ

*C*= ⊔

_{ξ}

*C*ξ with ξ ∈

*P*of form Eq. 3.5. We then put 3.9 where ω

_{0}is given by Eq. 3.6 and

*d*

_{ξ}is the

*d*-block in Eq. 3.5. We can prove that this is well defined and gives a ring-injection.

Given *x* ∈
*K*_{n}^{m}, we denote by
ν_{0}(*x*) the ideal of r which is the
inverse of the product of all the elementary divisor ideals of
*x* not contained in r; we put then
ν(*x*) = [r:ν_{0}(*x*)].
We call *x* primitive if rank(*x*) = Min(*m,
n*) and all the elementary divisor ideals of *x* are
r.

Proposition 2. *Given* ξ *as in Eq.
***3.5**, *suppose that both e and*
(δθ)^{−1} (*e* − 1) *have
coefficients in* r *if t > 0. Let
**a* = *g*^{−1}*h**with primitive *[*g**h*]
∈ *r*_{2r}^{r}*and
**gb* = *j*^{−1}*k with primitive *[*j**k*]
∈ *r*_{r+t}^{r}. *Then**where**we**take**j* = 1_{r}*if**t* = 0.

We now define a formal Dirichlet series 𝔗 by 3.10 This is a formal version of the Euler factor of Eq. 1.2 at a fixed nonarchimedean prime.

Theorem 1. *Suppose**that* δϕ ∈
*GL*_{n}(*r*); *put**p* =
[*g*:*g* ∩ *q*]. (*Thus**p* = *q**if**K* = *F*.) *Then**Here θ ^{i} = 1 if i is even; when i is odd,
θ^{i} is −1 or 0 according as d =
r or* d ≠ r.

This can be proved in the same manner as in ref. 2 by means of
*Proposition 2*.

Since we are going to take localizations of a global unitary group, we
have to consider *G*^{ϕ} =
*G*(*V*, ϕ) of Eq. 2.1 with
*V* = *K*_{n}^{1},
*K* = *F* × *F*, and ρ defined
by (*x, y*)^{ρ} = (*y, x*), where
*F* is a locally compact field of characteristic 0 with
respect to a discrete valuation. Let g and p
be the valution ring of *F* and its maximal ideal; put
r = g × g and *p*
= [g:p]. We consider ℜ(*C,
G*^{ϕ}) with *C* =
*G*^{ϕ} ∩
*GL*_{n}(r). Then the projection map pr
of *GL*_{n}(*K*) onto
*GL*_{n}(*F*) gives an isomorphism
η:ℜ(*C, G*^{ϕ}) →
ℜ(*E*_{1},
*GL*_{n}(*F*)), where
*E*_{1} = *GL*_{n}(g).
To be explicit, we have η(*C*(*x,
^{t}x*

^{−1})

*C*) =

*E*

_{1}

*xE*

_{1}. Let ω

_{1}denote the map of Eq. 3.6 defined with

*n*,

*E*

_{1}, and

*F*in place of

*r, E*, and

*K*. Putting ω = ω

_{1}○ η, we obtain a ring-injection 3.11 For

*z*= (

*x, y*) ∈

*K*

_{n}

^{n}with

*x, y*∈

*F*

_{n}

^{n}put ν

_{1}(

*z*) = ν(

*x*) and ν

_{2}(

*z*) = ν(

*y*), where ν is defined with respect to g instead of r. We then put 3.12 Then we obtain 3.13

## Section 4.

We now take a totally imaginary quadratic extension
*K* of a totally real algebraic number field *F* of
finite degree. We denote by **a** (resp. **h**) the set
of archimedean (resp. nonarchimedean) primes of *F*; further
we denote by g (resp. r) the maximal order of
*F* (resp. *K*). Let *V* be a vector space
over *K* of dimension *n*. We take a
*K*-valued nondegenerate ɛ-hermitian form ϕ on
*V* with ɛ = 1 with respect to the Galois involution of
*K* over *F*, and define *G*^{ϕ}
as in Section 2. For every *v* ∈ **a** ∪
**h** and an object *X*, we denote by
*X*_{v} its localization at *v*. For
*v* ∈ **h** not splitting in *K* and
for *v* ∈ **a**, we take a decomposition
4.1
with anisotropic θ′_{v} and a
nonnegative integer *r*_{v}. Put
*t*_{v} = dim(*T*_{v}). Then
*n* = 2*r*_{v} +
*t*_{v}. If *n* is odd, then
*t*_{v} = 1 for every *v* ∈
**h**. If *n* is even, then *t*_{v} =
0 for almost all *v* ∈ **h** and
*t*_{v} = 2 for the remaining *v* ∈
**h**. If *n* is odd, by replacing ϕ by
*c*ϕ with a suitable *c* ∈ *F*, we
may assume that ϕ is represented by a matrix whose determinant times
(−1)^{(n−1)/2} belongs to
*N*_{K/F}(*K*).

We take and fix an element κ of *K* such that
κ^{ρ} = −κ and
*i*κ_{v}ϕ_{v} has
signature (*r*_{v} + *t*_{v},
*r*_{v}) for every *v* ∈
**a**. Then
*G*(*i*κ_{v}ϕ_{v})
modulo a maximal compact subgroup is a hermitian symmetric space which
we denote by ℨ_{v}^{ϕ}. We take a suitable
point **i**_{v} of
ℨ_{v}^{ϕ} which plays the role of
“origin” of the space. If *r*_{v} = 0, we
understand that ℨ_{v}^{ϕ} consists of a single
point **i**_{v}. We put ℨ^{ϕ} =
∏_{v∈a}
ℨ_{v}^{ϕ}. To simplify our notation, for
*x* ∈ *K*_{A}^{×} or
*x* ∈
(**C**^{×})^{a}, *a*
∈ **Z**^{a}, and *c* ∈
(**C**^{×})^{a}, we put
4.2
For ξ ∈ *G*_{v}^{ϕ} and
*w* ∈ ℨ_{v}^{ϕ}, we define
ξ*w* ∈ ℨ_{v}^{ϕ} in a
natural way and define also a scalar factor of automorphy
*j*_{ξ}(*w*) so that
det(ξ)^{rv}*j*_{ξ}(*w*)^{−n}
is the jacobian of ξ. Given *k*, ν ∈
**Z**^{a}, *z* ∈ ℨ^{ϕ},
and α ∈ *G*_{A}^{ϕ},
we put
4.3
Then, for a function *f*:ℨ^{ϕ} →
**C**, we define
*f*∥_{k,ν}α:ℨ^{ϕ}
→ **C** by
4.4
Now, given a congruence subgroup Γ of
*G*^{ϕ}, we denote by
𝔐_{k,ν}^{ϕ}(Γ) the vector
space of all holomorphic functions *f* on ℨ^{ϕ}
which satisfy *f*∥_{k,ν}γ =
*f* for every γ ∈ Γ and also the cusp condition if
*G*^{ϕ} is of the elliptic modular type. We then
denote by 𝔖_{k,ν}^{ϕ}(Γ)
the set of all cusp forms belonging to
𝔐_{k,ν}^{ϕ}(Γ). Further, we
denote by 𝔐_{k,ν}^{ϕ} resp.
𝔖_{k,ν}^{ϕ} the union of
𝔐_{k,ν}^{ϕ}(Γ) resp.
𝔖_{k,ν}^{ϕ}(Γ) for all
congruence subgroups Γ of *G*. If ϕ is anisotropic, we
understand that 𝔖_{0,ν}^{ϕ} =
**C**.

Let *D* be an open subgroup of
*G*_{A}^{ϕ} such that
*D* ∩
*G*_{h}^{ϕ} is compact.
We then denote by
𝔖_{k,ν}^{ϕ}(*D*) the
set of all functions **f**:
*G*_{A}^{ϕ} →
**C** satisfying the following conditions:
4.5
for every
*p* ∈ *G*_{h}^{ϕ}
there exists an element
*f*_{p} ∈ 𝔖_{k,ν}^{ϕ}
such that
4.6
We now take *D* in a special form. We take a maximal
r-lattice *M* in *V* whose norm is
g in the sense of ref. 3 (p. 375) and put
4.7
4.8
4.9
where d is the different of *K* relative to
*F* and c is a fixed integral
g-ideal. Clearly *M̃* is an
r-lattice in *V* containing *M*, and we
easily see that *D*^{ϕ} is an open subgroup of
*G*_{A}^{ϕ}. We assume
that
4.10
Define a subgroup 𝔛 of
*G*_{A}^{ϕ} by
4.11
We then consider the algebra ℜ(*D*, 𝔛) consisting of
all the finite linear combinations of *D*τ*D* with
τ ∈ 𝔛 and define its action on
𝔖_{k,ν}^{ϕ} (*D*) as
follows. Given τ ∈ 𝔛 and **f** ∈
𝔖_{k,ν}^{ϕ}(*D*),
take a finite subset *Y* of
*G*_{h}^{ϕ} so that
*D*τ*D* =
⊔_{η∈Y}*D*η and define
**f**|*D*τ*D*:*G*_{A}^{ϕ}
→ **C** by
4.12
These operators form a commutative ring of normal operators on
𝔖_{k,ν}^{ϕ}(*D*).

For *x* ∈
*G*_{A}^{ϕ}, we define
an ideal ν_{0}(*x*) of r by
4.13
where ν_{0}(*x*_{v}) is defined as in
Section 3 with respect to an
r_{v}-basis of
*M*_{v}. Clearly ν_{0}(*x*)
depends only on *CxC*.

Let **f** be an element of
𝔖_{k,ν}^{ϕ}(*D*)
that is a common eigenfunction of all the *D*τ*D*
with τ ∈ 𝔛, and let
**f**|*D*τ*D* =
λ(τ)**f** with λ(τ) ∈ **C**. Given a Hecke
ideal character χ of *K* such that |χ| = 1, define a
Dirichlet series 𝔗(*s*, **f**, χ) by
4.14
where χ* is the ideal character associated with χ and
*N*(a) is the norm of an ideal a.
Denote by χ_{1} the restriction of χ to
*F*_{A}^{×}, and by θ the Hecke
character of *F* corresponding to the quadratic extension
*K*/*F*. For any Hecke character ξ of *F*,
put
4.15
From *Theorem 1* and Eq. 3.13, we see that
4.16
with a polynomial *W _{q} of degree
n* whose constant term is 1, where q runs over
all the prime ideals of

*K*prime to c. Let

*Z*(

*s*,

**f**, χ) denote the function of Eq. 4.16. Put 4.17 Theorem 2.

*Suppose*

*that*χ

_{a}(

*b*) =

*b*

^{μ}|

*b*|

^{iκ−μ}

*with*μ ∈

**Z**

^{a}*and*κ ∈

**R**

^{a}*such*

*that*∑

_{v∈a}κ

_{v}= 0.

*Put*

*m*=

*k*+ 2ν − μ

*and*

*with γ*

_{v}defined by*Then*ℜ(

*s,*

**f**, χ)

*can be continued to the whole s-plane as a meromorphic function with finitely many poles, which are all simple*.

*It is entire if*χ

_{1}≠ θ

^{ν}

*for*ν = 0, 1.

We can give an explicitly defined finite set of points in which the
possible poles of ℜ belong. Notice that *p*_{v} and
*q*_{v} are polynomials; in particular,
*p*_{v} = 1 if 0 ≤ *m*_{v} ≤
*k*_{v} and *q*_{v} = 1 if
|μ_{v} − 2ν_{v}| ≥
*n* − 1.

The results of the above type and also of the type of
*Theorem 3* below were obtained in refs. 2, 4, and 5 for the
forms on the symplectic and metaplectic groups over a totally real
number field. The Euler product of type *Z*, its analytic
continuation, and its relationship with the Fourier coefficients of
**f** have been obtained by Oh (6) for the group
*G*^{ϕ} as above when ϕ =
η_{r}.

## Section 5.

We now put (*W*, ψ) = (*V*,
ϕ) ⊕ (*H*_{m}, η_{m}) as in
Eq. 2.5 with (*V*, ϕ) of Section 4 and
*m* ≥ 0. Writing simply *I* =
*I*_{m}, we can consider the parabolic subgroup
*P*_{I}^{ψ} of
*G*^{ψ}. We put *P*^{ψ} =
*P*_{I}^{ψ} for
simplicity, λ_{0}(α) =
det(λ_{I}^{ψ}(*p*))
for *p* ∈ *P*^{ψ}, and
5.1
with *M* of Section 4 and the standard basis
{ɛ_{i},
ɛ_{m+n+i}}_{i=1}^{m}
of *H*_{m}. We can define the space
ℨ^{ψ} and its origin **i**^{ψ} in the
same manner as for *G*^{ϕ}. We then put
5.2
5.3
Here *e*_{v} is the element of
End(*V*_{v}) defined for *x*_{v} by
*wx*_{v} − *we*_{v} ∈
(*H*_{m})_{v} for
*w* ∈ *V*_{v}. We define an
**R**-valued function *h* on
*G*_{A}^{ψ} by
5.4
Taking **f** ∈
𝔖_{k,ν}^{ϕ}(*D*^{ϕ})
and χ as in Section 4, we define
μ:*G*_{A}^{ψ} →
**C** as follows: μ(*x*) = 0 if *x* ∉
*P*_{A}^{ψ}*D*^{ψ};
if *x* = *pw* with *p* ∈
*P*_{A}^{ψ} and
*w* ∈ *D*^{ψ} ∩
*C*_{0}^{ψ}, then we put
5.5
where χ_{c} =
∏_{v|c} χ*v*. Then we
define *E*(*x, s*) for *x* ∈
*G*_{A}^{ψ} and
*s* ∈ **C** by
5.6
This is meaningful if χ_{a}(*b*) =
*b*^{k+2ν}|*b*|^{iκ−k−2ν}
with κ ∈ **R**^{a},
∑_{v∈a} κ_{v}
= 0, and the conductor of χ divides c. We take such a
χ in the following theorem. The series of Eq. 5.6 is the
adelized version of a collection of several series of the type in Eq.
1.3.

Theorem 3. *Define γ _{v} as in*
Theorem 2

*with m = 0. Put*

*Then the product*

*can be continued to the whole*s-

*plane as a meromorphic function with finitely many poles*,

*which are all simple*.

We can give an explicitly defined finite set of points in which the possible poles of the above product belong.

## Section 6.

Let *G* be an arbitrary reductive
algebraic group over **Q**. Given an open subgroup *U*
of *G*_{A} containing
*G*_{a} and such that *U* ∩
*G*_{h} is compact, we put
*U*^{a} = *aUa*^{−1} and
Γ^{a} = *G* ∩
*U*^{a} for every *a* ∈
*G*_{A}. We assume that
*G*_{a} acts on a symmetric space 𝔚, and we
let *G* act on 𝔚 via its projection to
*G*_{a}. We also assume that
Γ^{a}/𝔚 has finite measure, written
vol(Γ^{a}/𝔚), with respect to a fixed
*G*_{a}-invariant measure on 𝔚. Taking a
complete set of representatives 𝔅 for
*G/G*_{A}/*U*, we put
6.1
where *T* is the set of elements of *G*
which act trivially on 𝔚, and we assume that
[Γ^{a} ∩ *T*:1] is finite. Clearly
σ(*U*) does not depend on the choice of 𝔅. We call
σ(*G, U*) the mass of *G* with respect to
*U*. If *G*_{a} is compact, we take
𝔚 to be a single point of measure 1 on which
*G*_{a} acts trivially. Then we have
6.2
We can show that σ(*U*′) =
[*U*:*U*′]σ(*U*′) if *U*′ is a
subgroup of *U*. If strong approximation holds for the
semisimple factor of *G*, then it often happens that both
[Γ^{a} ∩ *T*:1] and
vol(Γ^{a}/𝔚) depend only on *U*, so
that
6.3
If *G*_{a} is compact and *U* is
sufficiently small, then Γ^{a} = {1} for
every *a*, in which case we have σ(*U*) =
#(*G/G*_{A}/*U*). If *U* is
the stabilizer of a lattice *L* in a vector space on which
*G* acts, then
#(*G/G*_{A}/*U*) is the number of
classes in the genus of *L*. Therefore, σ(*U*) may
be viewed as a refined version of the class number in this sense.

Coming back to the unitary group *G*^{ϕ} of Section
4, we can prove the following theorem.

Theorem 4. *Suppose that
G*_{a}^{ϕ}*is compact*. *Let M be
a* g-*maximal lattice in V of norm*
g *and let* d *be the different
of K relative to F. Define an open subgroup D of
G*_{A}^{ϕ}*by Eq.
***4.9***with an integral ideal* c.
*If n is odd*, *assume that* ϕ *is represented
by a matrix whose determinant times*
(−1)^{(n−1)/2}*belongs to
N*_{K/F}(*K*); *if n is
even*, *assume that* c *is divisible by
the product* e *of all prime ideals for which
t _{v}* = 2.

*Then*

*where*

*d*= [

*F*:

**Q**],

*D*

_{F}is the discriminant of*F*,

*and*

*A*= 1

*or*

*A*=

*N*(

*e*)

^{n}

*N*(

*d*)

^{−n/2}

*according*

*as*

*n*

*is*

*odd*

*or*

*even*.

If *n* is odd, we can also consider σ(*D*′) for
6.4
with an arbitrary integral ideal c. Then
σ(*D*′) = 2^{−τ}σ(*D*), where τ is
the number of primes in *F* ramified in *K*.

## Section 7.

Let us now sketch the proof of the above
theorems. The full details will be given in ref. 7. We first take 𝔅
⊂ so that
=
⊔_{b∈𝔅}*G*^{ϕ}*bD*^{ϕ}.
Given *E*(*x, s*) as in Eq. 5.6, for each
*q* ∈
*G*_{h}^{ψ} we can
define a function *E*_{q}(*z, s*) of
(*z, s*) ∈ ℨ^{ψ} × **C** by
7.1
The principle is the same as in Eq. 4.6, and so it is
sufficient to prove the assertion of *Theorem 3* with
*E*_{q}(*z, s*) in place of
*E*(*x, s*). In particular, we can take *q*
to be *q* = *b* ×
1_{2m} with *b* ∈ 𝔅. Define
(*X*, ω) as in Eq. 2.5. Then there is an
isomorphism of (*X*, ω) to
(*H*_{m+n},
η_{m+n}) which maps
*P*_{U}^{ω} of
*Proposition 1* to the standard parabolic subgroup
*P* of *G*(η_{m+n}).
Therefore, we can identify ℨ^{ω} with the space
h^{a} with
7.2
We can also define an Eisenstein series *E*′(*x,
s*; χ) for *x* ∈
*G*_{A}^{ω} and
*s* ∈ **C**, which is defined by Eq.
5.6 with
(*G*(η_{m+n})_{A},
*P*, 1) in place of
(*G*_{A}^{ψ},
*P*^{ψ}, **f**). Taking
*E*′ and (*q, a*) ∈
*G*_{h}^{ω} (with *a* ∈
𝔅) in place of *E*(*x, s*) and *q*, we can
define a function
*E*′_{q,a}(z,
*s*) of (z, *s*) ∈
h^{a} × **C** in the same
manner as in Eq. 7.1. There is also an injection ι of
ℨ^{ψ} × ℨ^{ϕ} into
h^{a} compatible with the embedding
*G*^{ψ} × *G*^{ϕ} →
*G*(η_{m+n}). We put then
7.3
for every function *g* on
h^{a}, where δ(*w, z*) is a
natural factor of automorphy associated with the embedding ι. Take a
Hecke eigenform **f** as in Section 4 and define
*f*_{a} by the principle of Eq. 4.6. Then,
employing *Proposition 1*, we can prove
7.4
where *q* = *b* ×
1_{2m}, *A* is a certain gamma factor, and
Φ_{a} =
Γ^{a}/ℨ^{ϕ}. The computation is
similar to, but more involved than, that of ref. 4 (Section 4). Since
the analytic nature of *E*′_{q,a}
can be seen from the results of ref. 8, we can derive *Theorem
3* from Eq. 7.4.

Take *m* = 0. Then ψ = ϕ and
*E*_{q}(*z, s*) =
*f*_{b}(*z*). Then the analytic nature of
𝔗 (*s*, **f**, χ), and consequently that of
*Z*(*s*, **f**, χ), can be derived from Eq.
7.4. However, here we have to assume that
χ_{a}(*b*) =
*b*^{k+2ν}|*b*|^{iκ−k−2ν}
with κ ∈ **R**^{a},
∑_{v∈a} κ_{v}
= 0, and the conductor of χ divides c. The latter
condition on c is a minor matter, but the condition on
χ_{a} is essential. To obtain
*Z*(*s*, **f**, χ) with an arbitrary χ, we
have to replace *E*′_{q,a} by
𝔇*E*"_{q,a}, where *E*"
is a series of type *E*′ with 2ν − μ in place of
*k* and 𝔇 is a certain differential operator on
h^{a}.

As for *Theorem 4*, we take again ψ = ϕ and observe that a
constant function can be taken as **f** if
*G*_{a}^{ϕ} is compact. The space
ℨ^{ϕ} consists of a single point. The integral on the
right-hand side of Eq. 7.4 is merely the value
(*E*′_{q,a})°(*z, w*;
*s*). We can compute its residue at *s* =
*n* explicitly. Comparing it with the residue on the left-hand
side, we obtain *Theorem 4* when c satisfies Eq.
4.10. If *n* is odd, we can remove this condition
by computing a group index of type [*U*:*U*′].

- Copyright © 1997, The National Academy of Sciences of the USA

## References

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- Shimura G

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- Shimura G

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