44
M. L. RACINE
planes) and by the d 's corresponding to lines. To distinguish these d's
from those of the planes we will refer to them as elementary. Denote
H(ae.. +Ien) by a ^ , Hfae.^ by a
[ u ]
.
THEOREM 2. Let g = #(& , & , *), n 2, * induced by a hermitian
form, & an associative division algebra over a complete discrete valuation
field K. Any maximal order M of $ can be written M = ^nE(L) where L
is an D-lattice of 1/ such that L = L l L , L, i-modular. Conversely
M
= g nE(L), L = L ± L , L. i-modular, is a maximal order of g unless
& has no symmetric prime and L- is a (necessarily subnormal) plane.
The first statement follows from Proposition 2 and Theorem 1. The
second is a consequence of the following theorem.
THEOREM 3. Given an fl-lattice L of V with
/ d l C l \ / d 2 r - l C 2r-l \ ,
°1 V \
C
2r- l
d
2r /
such that l d
2
. _
1
I J d
2 i
| l ^ . ^ l . 1 i r. M= ^nE(L) is maximal if
and only if
(1) | v(d.) - v(d.)| 1 for all d,,d. elementary, all c ^ c ,
l j
I
j
Jo
m
(2) | v ( c ^ ) - v(cm)| 1
(3) | v(dt) - v(c^)| 1
unless there is one and only one with v(c«) odd and & has no symmetric
prime, in which case M is not maximal.
Note that if & has no symmetric prime v(&Q) = 2Z, so (1) becomes
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