1.3 7

v in S and the functions

Ψno)v (z

,v

(gv) over the places v outside S. Unfortunately,

the adelic Green function behaves too badly along H(A) for the Poincar´ e series

∑

γ∈H(F )\G(F )

Ψ(z)(n|s; γg) to be convergent absolutely. To compensate this defect,

we define its smoothing Ψβ,λ(n|s,g) by auxiliary introducing an even entire function

β(z) with fast decay as |Im(z)| → ∞ together with a complex parameter λ, and

by taking the contour integral of

Ψ(z)(n|s;

g) against β(z)/(λ + z) along a vertical

contour Re(z) = σ (see 6.4.1). For our purpose, the most important variant of

Green’s functions is the regularized smoothed kernel function defined as

ˆ

Ψ

β,λ

(n|α; g) =

1

2πi

#S

Re(s)=c

Ψβ,λ(n|s; g) α(s) dμS(s),

where α(s) is a certain even entire function in s, dμS(s) is a holomorphic form

on the complex manifold XS :=

v∈Σ∞

C ×

v∈S−Σ∞

(C/4πi(log qv)−1Z) and the

integration is the multidimensional contour integral (see 6.4.2).

For a cusp form ϕ on G(A) with trivial central character, the integral

Z(A)H(F )\H(A)

ϕ(h) η(h) dh (1.6)

is often called the (H, η)-period of ϕ, where Z denotes the center of G.

In §7, we introduce a regularization procedure for the (H, η)-period for ϕ not

necessarily cuspidal in such a way that the (H, η)-regularized period Preg(ϕ)

η

coin-

cides with the absolutely convergent integral (1.6) when ϕ is happend to be cus-

pidal. We explicitly compute the regularized periods for basic automorphic forms

constructed in §2 in terms of the associated L-functions (see Lemmas 7.4, 7.5 and

7.8). Although the computation is quite standard for cuspidal new forms ([27],

[5]), some extra work is necessary to treat old forms simultaneously.

In §8 and §9, we make the average of the kernel functions constructed in §6 over

the discrete orbit space H(F )\G(F ) to obtain the associated automorphic kernel

functions. The convergence of the infinite series pertaining to this procedure and

the square integrability of the resulting automorphic kernel functions are discussed

here. The crucial property of the automorphic Green function

Ψβ,λ(n|s; g) =

γ∈H(F )\G(F )

Ψβ,λ(n|s; γg), g ∈ G(A), s = (sv)v∈S ∈ XS

with suﬃciently large Re(sv) for all v ∈ S is established in Lemma 8.2, which is

a key to deduce the spectral expansion of the automorphic regularized smoothed

kernel

ˆ

Ψ β,λ(n|α; g) =

γ∈H(F )\G(F )

ˆ

Ψ β,λ(n|α; γg), g ∈ G(A) (1.7)

in 9.1. These functions depend holomorphically on an auxiliary complex parameter

λ, which should be kept large for the defining series to be convergent. (Other than

λ, the latter function also depends on an ideal n ∈ IS,η, ∗ an auxiliary entire function

β and a test function α on the space XS.) We continue the holomorphic function

(1.7) in λ meromorphically to a neighborhood of the point λ = 0, showing that

the constant term at λ = 0 is proportional to β(0); then we define

ˆ

Ψ reg(n|α) to be

the proportionality constant. The required analysis is carried out in the last part

of §9. In the course (Lemmas 9.7 and 9.9), we need a (weak) polynomial bound

of automorphic forms with both the spatial parameter and the spectral parameter