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.
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