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 c« 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