I. AUTOMORPHIC FORMS AND ABELIAN VARIETIES

p(x, y) + pf(x\ y') for x, y eV and x\ y' G V. We then view U(ip) x U{p') as a

subgroup of U(ip) in an obvious way.

1.2. We shall often express various objects by matrices. To simplify our nota-

tion, for a matrix x with entries in K we put

(1.5) z* = V \ X-P = {XP)~\ x =

tx~P,

assuming x to be square and invertible if necessary. Now let V = K^ and ip —

€(p* G K™. Then we can define an e-hermitian form (po on V by po(x, y) = x(py*

for x, y eV. In this setting we shall always write simply (p for the form (po. Then

we have

(1.6) GU(p) = { a G GLm{K) | apa* = v(a)p with i/(a) G F x },

(1.7) t % ) = { a G GL

m

(# ) | a^a* = ^ }, S £ % ) = £% ) n SLm(K).

We shall often consider U(r)n) with

" 0 - l

r

.In 0

Here we are taking e ~ — 1. In particular, if if = F, the group U(nn) is usually de-

noted by Sp(n, F). More generally, for a commutative ring A with identity element

we put

(1.9) Sp(n, A)= {ae GL2n(A) \

lanna

= rjn},

(1.10) Gp(n, A) = {ae GL2n{A) \

lar]na

= v{a)nn with I/(Q) G A

X

}.

Notice that

(1.11) det(a) =

v{a)n

(a G Gp(n, A)),

(1.12) det(a) det(a)' =

i/(a)m

(a G GU{(p)).

The latter formula is obvious. To prove (1.11), let a G Gp(n, A) and /? =

diag[ln, i/(a)l

n

]. Then /? G Gp(n, A) and i/(/3) = i/(a); thus

j3~la

G 5p(n, A). It

is well-known that det (Sp(n, A)) = 1, and hence det(a) = det(/3) = ^(a)

n

, which

is (1.11). In particular Gp(l, A) = GL2(-A) and 5p(l, A) = SL

2

(4).

a 6

(1.8)

^/n

Let £

d

G GL,2n{K) with a, 6, c, d of size n and let 5 G Fx. Then

(1.13) £eGU{7]n) and z/(f) = s ^ a*d-c*6 = sl

n

, a*c = c*a, b*d = d*b,

4= ad* - 6c* = sl

n

, ab* = 6a*, cd* = dc*.

1.3. Lemma, (1) Let A be a commutative ring with identity element. Let

x =

G GLm+n(i4) and a;

of size n. Then det (a;) det (e)

a 6

(2) If £

d

with a, e of size m and d, h

det(d).

G SU(r)n) with a, b, c, d of size n, then det (a), det (6),

det(c), and det(d) all belong to F.

(3) Every element ofGU(r]n) is a diagonal matrix times an element of SU(rjn).

(4) GU(p)/ [Fx U{ip)] is isomorphic to a subgroup of Fx /{ a2 | a G Fx }. Con-

sequently if A is a homomorphism of GU(p) into a group whose kernel contains

FxU{p), then A2 = l.

PROOF.

For the proof of (1) and (2), see [S97, Lemmas 2.15 and 2.16]. To prove

(3), let a G GU(r)n), p = v{a)~\ and (3 = diag[ln, pln]a. Then 0 G U{nn).