CHAPTER 1
Introduction
1.1
Let φ be a weight k holomorphic cuspform on the Poincar´ e upper half plane H
with respect to the modular group Γ0(N) of prime level N. In [41], Ramakrishnan
and Rogawski considered the average of central values for a product of the modular
L-series L(s, φ) and its quadratic twist L(s, φ η) divided by the Petersson norm
squared
φ
2
= [SL2(Z) :
Γ0(N)]−1
Γ0(N)\H
|φ(τ)|2 yk
dxdy
y2
for such φ’s :
φ∈Fk(N)new
ap(φ)∈J
L(1/2,φ) L(1/2,φ η)
φ 2
, (1.1)
where
Fk(N)new
is an orthogonal basis of the space of weight k( 4) Hecke-eigen
holomorphic cuspidal newforms for Γ0(N), and the summation is taken for those
φ such that its p-th normalized Hecke eigenvalue ap(φ) belongs to a given interval
J [−2, 2]. When the level N, the prime p and the conductor of the quadratic
Dirichlet character η are mutually co-prime, they obtained an asymptotic formula
as N +∞ of the average (1.1), observing that the Sato-Tate measure of J showed
up in the main term and explicating an error term. Moreover, they suggested a
similar kind of asymptotic formula for Maass cusp forms should be true. One of
our aims in this article is to realize this expectation in a setting of automorphic
representations of GL(2) over a totally real algebraic number field. We remark that
the result of [41] is generalized to holomorphic Hilbert modular case by [10].
To explain the main result about this theme, let us introduce some notation
which will be used in this article throughout. Let F be a totally real number field,
dF = [F : Q] its degree and oF its maximal order. Let Σfin and Σ∞ be the set of
finite places of F and the set of infinite places of F , respectively. The completion of
F at a place v Σfin Σ∞ is denoted by Fv. If v Σfin, Fv is a non-archimedean
local field, whose maximal order is denoted by ov; we fix a uniformizer
v
of ov
once and for all, and designate by qv the order of the residue field ov/pv, where
pv = vov is the maximal ideal of ov. Let A and Afin be the adele ring of F and
the finite adele ring of F , respectively. For an oF -ideal a, let us designate by S(a)
the set of all v Σfin such that aov pv.
Let S be a finite set of places containing Σ∞, and η =
v∈Σ∞∪Σfin
ηv an
idele-class character of order 2 with conductor f such that η is unramified over
Sfin = S Σfin and such that ηv is trivial for any v Σ∞. Let IS,η

be the set of all
the square free oF -ideals n with the following properties.
1
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