INTRODUCTION

5

in G^, and G^ has a parabolic subgroup whose reductive factor is G* x GLq(K).

Given a suitable congruence subgroup JT" of G^, we can define an Eisenstein series

E{z, s; f, x) for (z, s) G 3 ^

x

C with respect to that parabolic subgroup and the

set of data (f, x -O- Now

w e

easily

s e e

that diag[?/, —p] is equivalent to rjn+q,

so that G^ x G^ can be embedded into G{r)n+q), and 3^ x 3^ can be embedded

into Hn+q. Pulling back an Eisenstein series on Hn+q of split type to 3^ x 3^, we

obtain a function H(z, w\ s) of (z, w; s) G 3^ x 3^ x C, with which we proved in

[S97] an equality that takes the form

(0.11) c(s)Z(s, f, x)E(* «; f, X) = A(s) / H{z, w;

s)f(w)6(w)mdw

in the simplest case, where f is a congruence subgroup of G^, c is an easy product

of gamma functions, A is a product of some L-functions, dw is a

G9-invariant

measure on 3^, and

6(w)rn

is a factor, similar to

yk

in the one-dimensional case,

that makes the integral meaningful. If \j) = p, then (0.11) takes the form

(0.12) c\s)Z{s, f, x)f (*) = A'(s) f H\z, w;

s)f(w)6(w)mdw.

We evaluate (0.11) and (0.12) at s = ao for ao belonging to a certain "critical

set," and observe that H(z, w; cr0) is nearly holomorphic in (z, w) G 3^ x 3*\ and

even Q-rational up to a power of n and a factor q as in (0.3). Then we can show

that

A(a0)H{z, w\ a0) =

waqY^gi{z)hi(w)

i

with some a G Z, and functions gi on 3^ and hi on 3^» which are nearly holo-

morphic and Q-rational. The same is true for A'if7; both gi and /i^ are defined

on 3^ then. This fact applied to (0.12) produces a proportionality relation

Z(70, f, x ) € 7 r ^ q ( p , , f ) Q

with some /? G Z and a Q-rational nearly holomorhic p'. Now we can show that

Z{s, f, x) ^ 0 for Re(s) 3^/2 if G = G(7y9) and for Re(s) n if G = G^ with y

of a general type. There is one more crucial technical fact that we can replace p' by

a Q-rational holomorphic cusp form p that belongs to the same Hecke eigenvalues

as f. Choosing a0 so that

Z(CF0,

f, x) ¥" 0? we can show that (p, f) / (f, f) G Q,

and eventually (0.3) for 7o belonging to an appropriate set. Strictly speaking,

(0.12) is true only under a consistency condition on (f, x) and the proof of (0.3)

in the most general case is more complicated.

Next, we evaluate (0.11) at a critical CTQ in a similar way, to find that

Z(s

0

, f, x)E(z, cro; f, x) = ^ q (r, f) g(z)

with some Q-rational nearly holomorphic function g on 3 ^ and some r of the

same type as the above p. Dividing this equality by (f, f) and employing (0.3),

we obtain the desired near holomorphy and Q-rationality of E(z, ao; f, x) which

is the final main result of this book.

Since the title of each section can give a rough idea of its contents, we shall not

describe them here for every section. However, there are some points which are not

discussed in the above, and on which special attention may be paid. Let us note

here some of the noteworthy aspects.