right definition. To explain about this point, we first note that J T \ 3 ^ has a structure
of algebraic variety that has a natural model W defined over Q. We call then a
automorphic function (that is, T-invariant meromorphic function on 3^ satisfying
the cusp condition) Q-rationai (or arithmetic) if it corresponds to a Q-rational
function on W in the sense of algebraic geometry. Now there are two basic facts:
(0.5) The value of a Q-rational automorphic function at any CM-point of3^, if
finite, is algebraic.
(0.6) If f and g are Q-rational automorphic forms of the same weight, then f/g
is a Q-rational automorphic function.
Here a CM-point on y is defined to be the fixed point of a certain type of torus
contained in G. If G = G71 and q = 1, then H is the standard upper half plane,
and any point of H belonging to an imaginary quadratic field is a CM-point and
vice versa. In such a special case, (0.5) and (0.6) follow from the classical theory of
complex multiplication of elliptic modular functions. In more general cases, (0.5)
was established by the author in the framework of canonical models. As for (0.6),
it makes sense if G = G71, and we can indeed give a proof, if nontrivial, of (0.6) in
such a case. For cp of a more general type, however, (0.6) is a meaningful statement
only when we have defined the Q-rationality of automorphic forms. Thus it is one
of our main tasks to define the notion so that (0.6) holds.
Turning our eyes to Eisenstein series, easily posable questions are as follows:
(i) Assuming that E(z, 70) is finite, is E(z, 70) as a function of z holomorphic?
(ii) If that is so, is it Q-rational up to a well-defined constant?
Here we take meromorphic continuation of E(z, s) to the whole s-plane, as we
proved in [S97], into account. Every researcher of automorphic forms should be
able to accept such questions naturally, since the answers to them for G SL2(Q)
are well-known and fundamental. There is a marked difference between the Q-
rationality here and the arithmeticity of
since the latter concerns cro in
an interval which can be large, while E(z, do) can be holomorphic in z only at a
single point CTQ. NOW the interval, or rather the set of critical points belonging to
the interval, is suggested by the functional equation for Z, and we can find such a
set even for E(z, s) by means of its analytic properties. We cannot expect E(z, a0)
to be holomorphic in z for every critical point 70 in the set. We should also note
a classical example in the elliptic modular case:
Urn Y, {cz +
S _ 0#(c,d)GZ 2
oo ,
= ( 4 7 r y ) -
- 1 2 -
+ 2 ^ f ^ a
n = l ^
This is a nonholomorphic modular form of weight 2, and there are similar non-
holomorphic forms of weight (n -f 3)/2 with respect to a congruence subgroup of
Sp(n, Z). Therefore our next questions are:
(iii) What is the analytic nature of these E(z, cr0) ?
(iv) Can we still speak of the Q-rationality of such E(z,
One of the main purposes of this book is to answer these questions, which are not
only meaningful by themselves, but also closely connected with the arithmeticity of
I 2-Kinz
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