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
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