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)

(ι ∈ Σ∞). Here, Lfin(s, χ) =

v∈Σfin

(1−χv( v) qv

−s)−1

is the L-series of χ.