4 1. INTRODUCTION
j
uj(−x + iy) with some
j
{±1} for each j. Then, the L-series of uj is defined
to be the absolutely convergent series
Lj(s) =

n=1
cj(n)
ns
, Re(s) 0,
where

n∈Z
cj(n)
y1/2
Kνj
/2
(2π|n|y)
e2πinx
is the Fourier expansion of uj(x + iy)
at the cusp i∞. As is well-known, the completed L-series
ˆ
L
j
(s) = ΓR(s−νj) ΓR(s+
νj) Lj(s) is continued meromorphically to C with the functional equation
ˆ
L j(1−s) =
j
ˆ
L j(s). The asymptotic behaviours of various spectral means of the central values
Lj(1/2) are extensively studied by Motohashi [37] by means of Kuznetsov’s formula.
Among other things, the asymptotic formula of the square mean values
κj t
|Lj(1/2)|2
cosh(πκj)
=
2
π2
t2
(log t + CEuler 1/2 log(2π)) + O(t(log
t)6),
t +∞
(1.4)
is obtained ([37, Theorem 2]), where κj = iνj/2 and CEuler is the Euler con-
stant. In this paper, we prove an analogous asymptotic formula not for the mean
of
|Lj(1/2)|2/cosh(πκj)’s,
but rather for the mean of
|ˆj(1/2)|2’s
L in the context
of automorphic representations of GL(2,F ). This formula also can be viewed as
an analogue of multidimensional Weyl’s law for tempered cuspidal multiplicities of
automorphic representations ([9], [30]).
Theorem 1.3. Let n be any square free ideal. Let η be a real-valued idele
class character of F × unramified over n such that ηι(−1) = 1 for all ι Σ∞ and
v∈S(n)
ηv( v) = 1. Let J XΣ∞
0
be a compact subset with smooth boundary,
which is “positive”, i.e., Im(νι) 0 for any ν J and for any ι Σ∞. Then, for
any 0,
π∈Πcus(n)
νπ,Σ∞ ∈tJ
wn
η
(π)
L(1/2, π) L(1/2, π η)
N(n) L(1, π; Ad)
=
4(1 + δn,oF )
D3/2
F
vol(J)
(2π)dF
tdF
(dF R(η) log t +
Cη(F,
n))
+ O
tdF −1
(log
t)3
+ O
tdF (1+4θ)+
, t +∞,
where tJ = {tν| ν J },
wn(π)
η
=
N(nfπ
−1)
[K0(fπ) : K0(n)]
v∈S(nfπ
−1)
1 + ηv( v)
1 +
(qvv/2 ν
+ qv
−νv/2 )/(qv/2 1
+ qv
−1/2
)
,
Cη(F,
n) = CTs=1L(s, η) +
dF
2
(CEuler + 2 log 2 log π) +
log(N(n)1/2
DF ) R(η)
with L(s, η) =
ΓR(s)dF
v∈Σfin
(1 ηv( v) qv
−s)−1
the completed L-function of η,
R(η) = Ress=1L(s, η) and θ R is any constant satisfying
|Lfin(1/2 + it, χ)| = O(
ι∈Σ∞
(1 + |t +
b(χι)|)1/4+θ),
t R (1.5)
uniformly for all the unramified idele-class characters χ of F
×
with the archimedean
components χι(x) =
|x|ib(χι)
Σ∞). Here, Lfin(s, χ) =
v∈Σfin
(1−χv( v) qv
−s)−1
is the L-series of χ.
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