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