4

INTRODUCTION

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) =

p(H(z))-1D[p(E(z))g(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

2

(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

simplicity:

(0.10) If A is of total degree p in terms of d/dzij, then

TT~PA

preserves near

holomorphy and (^-rationality.

If G =

G77,

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

(n~~pAf)/g

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

G71

can be embedded