scalar-valued, but in order to use differential operators effectively, it is necessary to
consider vector-valued forms. If [K : Q] = 2 and G = G(r]q), such a form is defined
relative to a representation {p, X} of a group
& = { (a, 6) G GL9(C) x GLq(C) | det(a) = det(6) },
where X is a finite-dimensional complex vector space and p is a rational rep-
resentation of ^ into GL(X). Put T = C^ and view it as a global holomor-
phic tangent space of Hq; define a representation {p ® r, Hom(T, X)} of ^ by
[(p® r)(a, b)h] (u) = p(a, b)h{laub) for (a, 6) G J?, h G Hom(T, X), and ueT. For
a function # :H X we define Hom(T, X)-valued function £)# and Z}p# on W by
(^s)(«) = EL=i ««W*« e r),
(Dpg)(z) =
where z = {zij)(fj=1 G H and H : H £ is defined by S"(z) = (i(z—*z), i(z* z)).
These can also be defined on 3^ for (p of a general type. Then we can show that
if g is an automorphic form of weight p, then Dpg is a form of weight p ® r. If
7=1, then H is the standard upper half plane, G7? n SL2(K) = SL
(Q), £ = C x ,
p(a) = a*1 for a G C x with A : G Z, and H(z) = (2y, 2y) where y = Im(z); we can
easily identify D# with dg/dz, so that D^g = y~k(d/dz)(ykg), and (p 0 T)(G) =
flfc+2 Thus Dp is the well-known operator that sends a form of weight k to a form
of weight k + 2.
Now iteration of operators of this type, such as Dp®TDp, produces an automor-
phic form with values in a representation space of ^ of a large dimension if q 1,
even if we start with X C. Decomposing the space into irreducible subspaces and
looking particularly at the irreducible subspaces of dimension one, we can define
a natural differential operator A that sends scalar-valued automorphic forms to
scalar-valued forms of increased weight. The significance of these iterated opera-
tors and A are explained by the following fact, which is formulated only for A for
(0.10) If A is of total degree p in terms of d/dzij, then
preserves near
holomorphy and (^-rationality.
If G =
this can be derived from our definition in terms of expression (0.8). Now
property (0.9), if true, would imply that for a Q-rational holomorphic automorphic
forms / and g such that Af and g have the same weight, the value of
at any CM-point, if finite, is algebraic. This is highly nontrivial, and in fact we
first prove this special case of (0.9), and derive the general case from that result.
As for problem (3), we first investigate the Fourier expansion of E(z, s) of split
type. In fact, this was done in [S97], but here we examine the behavior of the
Fourier coefficients at a critical value of 5. Employing their explicit forms, we find
that E(z, a) is holomorphic in z and Q-rational, or is of the type (0.7), if the
weight of E and a belong to certain special types. For a more general weight and
a general 70, we prove that cE(z, ao) = AEf(z, a) with a suitable A, a nonzero
constant c, and a suitable E' belonging to those special types. Then (0.10) settles
problem (3) for E(z, a0).
To treat problems (4) and (5), let us now go back to the Euler product Z(s, f, x)
of (0.3) on G^; we refer the reader to [S97] for its precise definition. We consider
G^ with ip = diag[/?, 77], where 7 7 is as in (0.2). Then G^ x
can be embedded
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