10 I. AUTOMORPHIC FORMS AND ABELIAN VARIETIES

map of F'A into F

A

derived from the map Tr//F : F' — F when F ' is an algebraic

extension of F.

Let W be a finite-dimensional vector space over F, and L a ^-lattice in W.

Taking an element a of Wh, we define a function A on W^ by X(x) = Ylveh ^v{xv)

for x G Wh, where \v is the characteristic function of the coset Lv + av. We

then denote by S(Wh) the vector space of all finite C-linear combinations of such

functions A for all possible choices of (L, a). This is called the Schwartz-Bruhat

space of Wh- We view every ^€«S(Wh) as a function on WA by putting £(x) = ^(#h)

for x G WA- In particular, £(£) is meaningful for every £ £ W. We can easily see

that the restriction of the elements of 5(Wh) to W gives an isomorphism of S(W\l)

onto the set of all finite C-linear combinations of functions, each of which is the

characteristic function of a coset of W modulo a g-lattice in W. This is because

WA = W + Y with Y = {y eWA\yhe Y\veh Lv } for any fixed ^-lattice L.

We now put

(1.14) e(z) =

e27riz

(zGC) ,

and define characters eA " FA — T and ev : Fv —• T for each v G v as follows:

if v G a, then ev(x) = e(x) for real v and ev(x) = e(x + x) for imaginary v;

if v e h and p is the rational prime divisible by v, then e«(a;) = ep(Tri?v/Qp(x)),

where ep(z) = e(—y) with ?/ € Um=i^~ m ^ s u c n that z — y £ Zp. We then put

eA(^) = ELev e . f ^ ) , eh(x) = eA(^h), and ea(x) = e

A

(^

a

) for x £ FA. We note

here a basic property of ev :

d{F/Q)~l

= {xeFv\ ev(xy) = 1 for every ye$v} [v e h),

where D(F/Q) denotes the different of F relative to Q.

We insert here an easy fact as an application of Lemma 1.3 (4):

(*) For every a G GU(p)A the map x »-

OLXOL~1

of U(ip)A onto itself leaves

any fixed Haar measure of U(cp)A invariant.

Indeed, let \x be a Haar measure of U{ip)v for a fixed v G v. Then, for a G

GU((f)v we have ^(aXa'1) = X(a)fi(X) for every measurable set X in U(ip)v with

a positive real number A(a). Clearly A is a homomorphism of GU((p)v into R

x

and

A(FVX)

= 1. Also, \(U(if)v) = 1 by [S97, Proposition 8.13 (1)]. Therefore, by

Lemma 1.3 (4) we have A (a) = 1, since A (a) 0.

1.7. Let A be a principal ideal domain and F the field of quotients of A. We

call an element X of A™ primitive if rank(X) = Min(ra, n) and the elementary

divisors of X are all equal to A. If m = n, clearly X is primitive if and only if

X G GLn(A). If m n (resp. m n), then X is primitive if and only if X is the

first m rows (resp. n columns) of an element of GLn(A) (resp. GLm(A)). (For

these and other properties of primitive matrices, see [S97, Lemmas 3.3 and 3.4].)

Given x eF™, we can find c € ^ a n d d G GLm{F) DA™ such that [c d] is

primitive and x =

d~lc.

We then call the last equality a (left) reduced expression

for x, and define an integral ideal vo(x) by

(1.15) u0{x) = det(d)A.

This is independent of the choice of c and d. We call VQ{X) the denominator ideal

of x. We easily see that

ISQ(X

+ a) = ^o(#) if a - A (see Notation). For these and

other properties of the symbol I/Q see [S97, Proposition 3.6, §3.7, and Lemma 3.8].