8 1. INTRODUCTION

varied. Fortunatly, such a bound is established in a quite general setting in [13]

(for maximal cuspidal Eisenstein series). We quote the result in our setting in the

Appendix (Propositions 15.2 and 15.1).

In §10, we compute the regularized (H, η)-period Preg(

η

ˆ

Ψ reg(n|α)) using the

spectral expansion of

ˆ

Ψ reg(n|α) given by (9.14). The final outcome of §10 is Theo-

rem 10.5.

In §11 and §12, we compute Preg(

η

ˆ

Ψ reg(n|α)) with the defining series (1.7) subdi-

vided according to the double cosets H(F )γH(F ). Having convergence results to be

established in §11 which is the most technical section in this article, we explicitly

compute the regularized (H, η)-period of

ˆ

Ψ

reg

(n|α) according to the above men-

tioned subdivision by double cosets. The final result is given in Theorem 12.1. By

equating the two different expressions of Preg(

η

ˆ

Ψ reg(n|α)) given by Theorems 10.5

and 12.1, we arrive at the relative trace formula (13.1).

In §13, we prove Theorem 1.1 and Corollary 1.2 under a slightly weaker as-

sumption on η. In §14, imitating the technique used in [37], [9] and [30], we prove

Theorem 14.1, from which Theorem 1.3 follows immediately.

Acknowledgement

The author would like to thank Shingo Sugiyama, who read the manuscript

very carefully pointing out various mistakes and inaccuracies therein. Thanks are

also due to Masatoshi Suzuki for informing the author of the work [32] when it was

a preprint.

Notation : The number 0 is included in the set of natural numbers: N = {0, 1, 2,... }.

We set N∗ = N − {0}.

For two non-negative real-valued functions f(x) and g(x) on a set X, we write

f(x) g(x), x ∈ X (or f(x) = O(g(x))) if there exists a positive constant C

such that the inequality f(x) C g(x) holds for any x ∈ X. If f(x) g(x) and

g(x) f(x), we write f(x) g(x).

If a condition P is given, the Kronecker symbol δ(P) in a generalized sense is

defined by requiring that δ(P) is 1 if P is true and is 0 otherwise.

For any set X and its subset S, the characteristic function of S is denoted by

χS; thus χS(x) = δ(x ∈ S) for any x ∈ X.

For c ∈ R, let Lc denote the vertical line {s ∈ C| Re(s) = c } on the complex

plane; when we regard it as a contour, we give it the direction with increasing

imaginary part.