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 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) =
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