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).
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