2.7. EISENSTEIN SERIES AND THEIR L-FUNCTIONS 19

Proof. If ν ∈ iR, the hermitian inner product on Iv(χv||v

ν/2

) is Gv-invariant.

Using the easily confirmed relation

1 − Q(Iv(χv||v

ν/2))2

= χv(

v

)qv

−ν/2(1

+ qv

−1)

L(ν + 1,χv)

2

−2

, ν ∈ iR,

we infer the first statement from Lemma 2.2. The formula (2.25) is proved in

[5, Proposition 4.6.7] when dv = 0; the general case is similar. By (2.9), we see

that Mv(ν) sends the vector f1,χv

(ν)

to the vector f

(−ν)

1,χv

−1

multiplied by the same

constant occurring in (2.25). Then, taking the modification factor in (2.24) into

account, we obtain (2.26).

2.7.1. Fix χ. A family of smooth functions f

(ν)

∈ I(χ||A

ν/2

) with varying

ν ∈ C is said to be a flat section if f

(ν)|K

is independent of ν. A flat section

f

(ν)

∈ I(χ||A

ν/2

) is K-finite if R(K)f

(0)

spans a finite dimensional space in I(χ).

For such f

(ν),

the Eisenstein series is initially defined by the absolutely convergent

series

E(f

(ν);

g) =

γ∈BF \GF

f

(ν)(γg),

g ∈ GA, Re(ν) 1.

For each g ∈ GA, the holomorphic function ν → E(f

(ν);

g) on Re(ν) 1 is contin-

ued meromorphically to the whole complex plane, holomorphic on the imaginary

axis. The unique singularity of E(f

(ν);

g) on the half-plane Re(ν) 0 is a possible

pole at ν = 1, which occurs only when

χ2

= 1.

2.7.2. Let n be a square free oF -ideal. Then, the invariant part I(χ||A

ν/2

)K0(n)K∞

is non zero only when fχ = oF ; we assume this from now on. For each oF -ideal c di-

viding n, let

fχ,c) (ν

∈ I(χ||A

ν/2

) be the function coming from the decomposable tensor

{

v∈S(c)

˜(ν)

f

1,χv

}⊗{

v∈S(c)

f0,χv

(ν)

} by the isomorphism I(χ||A

ν/2

)

∼

=

v

Iv(χv||v

ν/2

).

Lemma 2.9. The family of functions

fχ,c) (ν

∈ I(χ||A

ν/2

) (ν ∈ C) is flat. When

ν ∈ iR, the set of functions

{fχ,c)| (ν

n ⊂ c } is an orthonormal basis of the invariant

part I(χ||A

ν/2

)K0(n)K∞

.

Proof. For any place v, the restriction f0,χv

(ν)

|Kv is identically 1. If v ∈ Σfin,

it turns out that

˜(ν)

f

1,χv

(k) equals

qv/2 1

or −qv

−1/2

according to k ∈ K0(pv) or k ∈

Kv − K0(pv), respectively. From these observations, the first assertion is evident.

Since the inner product (2.23) is a tensor product of inner products on local

spaces Iv(χv||v

ν/2

) which contains a unit vector

f0,v) (ν

at all places v, the second

assertion follows from the first statement of Lemma 2.8.

From now on, we abbreviate

E(fχ,c); (ν

g) to Eχ,c(ν; g). Let

Eχ,c(ν;

◦

g) =

F \A

Eχ,c(ν;[ 1 x

0 1

] g) dx

be the constant term of Eχ,c(ν; g). Then,

Lemma 2.10.

Eχ,c(ν;

◦

g) =

fχ,c)(g) (ν

+ DF

−1/2

Aχ,c(ν)

L(ν,

χ2)

L(ν + 1,χ2)

f

(−ν)

χ−1,c

(g), g ∈ GA,