ARITHMETICS OF JORDAN ALGEBRAS 31

Since M p and M are maximal orders, as was noted above (^nE(L.A)) 1 =

E(LA) and ( ^ n E ( L J ) ' = E(L

2

). Therefore E(L A) = E(L2) which implies

L A = QL (by Lemma 4) for a an o-ideal, so a L A = L . We have

-1 -1 -2 1 -2 -2

a

2

= g(L

2

,L

2

) = g(a LjA, a LjA) = a gfL^LjAA ) = o g(L

] [

,L

1

7) = 7 a a}

- 1 2

Hence 0 = 7 0 a and

\ Q

1

0 /

\ Q

2

0 /

q . e . d .

REMARK 2. If Q is trivial there is only one isomorphism clas s of

maximal orders of # = U(C ), C split.

COROLLARY 1. An order M of ? = M ( C J ) n 3 , C a split

n 7 —

associativ e composition algebra over K, is maximal if and only if M' is a

maximal order of C (a fortiori a maximal J -stabl e order of C ).

n 7 n

§4. Maximal ^-stabl e orders in associativ e algebras with involution over a

complete discrete valuation field.

Let ($, -) be a finite dimensional central associativ e division

algebra with involution over the field § (i. e. $ = W(C(&), - ) , C($) the

center of $). A hermitian form is a mapping h : \s x V -* & such that

(i) h(ax, y) = ah(x, y) a

€

fl, x, y

€

U.

(ii) h(y, x) = h(x, y).

Accordingly we say that A = (a..)

e

& is hermitian if a.. = a... A

U n i; 31

skew-hermitian form h is a mapping h : V x 1/ -* $ such that