1.2 3

Then, our first main result is stated as follows.

Theorem 1.1. As N(n) → +∞ with n ∈ IS,η,

∗

the measure λS(n)

η

converges

∗-wealky to the measure λS,

η

i.e.,

λS(n),f

η

→ λS,f

η

for any f ∈ Cc

0(XS 0+).

As a corollary to this theorem, we have

Corollary 1.2. Let {Jv}v∈S be a family of intervals such that Jv ⊂ [1/4, +∞)

if v ∈ Σ∞ and Jv ⊂ [−2, 2] if v ∈ Sfin. Then, for any δ 0, there exists an irre-

ducible cuspidal automorphic representation π with trivial central character having

the following properties.

(1) The conductor fπ of π belongs to IS,η

∗

and N(fπ) δ.

(2) L(1/2,π) = 0 and L(1/2,π ⊗ η) = 0.

(3) The spectral parameters νπ,S = (νv)v∈S at S of π satisfies (1 − νv 2)/4 ∈ Jv

for any v ∈ Σ∞ and qv

−νv/2

+

qvv/2 ν

∈ Jv for any v ∈ Sfin.

If we take a function φ on

HdF

corresponding to the new vector ϕπ

new

on

GL(2, A), then (1 − νv

2)/4

coincides with the v-th Laplace eigenvalue for φ and

qvv/2 ν

+ qv

−νv/2

coincides with the v-th Hecke eigenvalue for φ. Thus, Corollary 1.2

is regarded as an analogue of [41, Corollary B]. We deduce Theorem 1.1 and Corol-

lary 1.2 in the last part of §13 from a more general result Theorem 13.17.

Here are some comments on existing works related to our result. Theorem 1.1

means that the Satake parameters of automorphic representations weighted by the

central L-values are equidistributed in the parameter space. This kind of equidistri-

bution phenomenon of the Satake parameters (or Hecke eigenvalues) of automorphic

forms is first observed by Serre ([46]) and independently by Corney-Duke-Farmer

[7] for the elliptic modular forms without the weighting by L-values. A convergence

result for spectral average with L-value weighting slightly different from ours was

established by Royer ([45]) for the elliptic modular case, and by [41] and [10] for

the holomorphic Hilbert modular case with the same L-value weighting as ours.

Recently, Shin ([47]) far extended the Serre’s result to the cuspidal automorphic

representations whose archimedean components belong to a fixed discrete series L-

packet, working on a general reductive group by means of Arthur’s trace formula

applied to the Euler-Poincar´ e functions.

In somewhat different but related context, Trotabas ([52]) studied the first two

mollified moments of L-values L(1/2,π) for cuspidal representations π correspond-

ing to holomorphic Hilbert modular forms. In this work, the central L-values are

captured by the Dirichlet series expression (rather than the Euler product expres-

sion as we do) and the main tool of investigation is Petersson’s formula for Fourier

coeﬃcients extended to the general Hilbert modular setting.

1.2

To explain our second result, we start with the classical situation again. Let

=

−y2(∂2/∂x2

+

∂2/∂x2)

be the hyperbolic Laplacian acting on

L2

(

SL2(Z)\H;

y−2

dxdy

)

, and {λj = (1−νj

2)/4|

j 1 } the non-decreasing sequence

of the cuspidal eigenvalues of , counted with multiplicity. Fix an orthonormal

system of Maass cusp forms {uj}j=1

∞

such that uj = λj uj and uj(x + iy) =