Z(ao). In fact, the answers to (iii) and (iv) are indispensable for the proof of (0.3)
as we shall explain later, but first let us describe our answers.
We first define the notion of nearly holomorphic function on any complex mani-
fold with a fixed Kahler structure. Without going into details in the general case,
let us just say that a function on such a manifold 3 is called nearly holomorphic
if it is a polynomial of some functions n , ..., rm on 3, determined by the Kahler
structure, over the ring of all holomorphic functions on 3- If 3 is the above H
of tube type with a G-invariant Kahler structure, then the ri are the entries of
(z* — z ) - 1 , where z is a variable matrix on H. If 3 is a hermitian symmetric space,
there is also a characterization of such functions in terms of the Lie algebra of the
transformation group on 3-
Now we can naturally define nearly holomorphic automorphic forms by replacing
holomorphy by near holomorphy in the definition of automorphic forms. If G =
then such a form / on H has an expansion
(0.8) f(z) = EhPhiM** - z)}-1) exp (27Ti . tr(hz)) (z e H),
*s t n e s a m e
(0-4) and PhiX) is a polynomial function in the entries
of Y whose degree is less than a constant depending on /. We say that / is
M-rational if ph has all its coefficients in a field M for every h. For example,
the function of (0.7) is a Q-rational nearly holomorphic modular form. We shall
show that E(z, do) is indeed nearly holomorphic and Q-rational in this sense, up
to a constant, which is a power of TT if G =
Moreover, here is a noteworthy
consequence of our definition:
(0.9) If f and g are Q-rational nearly holomorphic automorphic forms of the same
weight, then the value of f/g at any CM-point ofH, if finite, is algebraic.
It should be noted that this is anything but a direct consequence of (0.6). Also, for a
general type of p we cannot use (0.8). However, once we have the Q-rationality of
holomorphic automorphic forms, we can at least define the Q-rationality of nearly
holomorphic automorphic forms by property (0.9), though it is of course nontrivial
to show that such a definition is indeed meaningful. So far we have taken G to be
unitary, but the symplectic case can be handled too; in fact it is similar to and easier
though the case of half-integral weight requires special consideration.
Having thus presented our problems in rough forms, we can now set our program
(1) We first define the Q-rationality of automorphic forms so that (0.6) holds.
(2) We define nearly holomorhic automorphic forms and their Q-rationality so
that (0.9) holds. _
(3) We prove the near holomorphy and Q-rationality of E(z, &o) up to a power of
ix in the easiest cases, namely, when G is symplectic or of type G77, and E is defined
with respect to a parabolic subgroup whose unipotent radical is a commutative
group of translations on H. Let us call such an E a series of split type.
(4) We prove (0.3) by using the result of (3).
(5) Finally we prove the near holomorphy and Q-rationality of E(z, ao) up to a
well-defined constant in the most general case.
Let us now briefly describe the technical aspect of how these can be achieved.
One important point is that certain differential operators on H are essential to
(2) and (3). In the above we tacitly assumed that our automorphic forms are