58 M. L. RACINE

fore, in general, even when & generates $, the o-algebra generated by

d(&0 nJJC{f v ( d ) , -)) need not be ©.

The following proposition and corollary can be proved by the methods

used to prove Theorems 2 and 3 and are included becaus e in the cas e of

algebraic number fields, skew-hermitian quaternionic forms are usually

considered.

PROPOSITION 7. Let ? = M(C , *), n 2, C a division quaternion

algebra over a local field K of characteristic not 2, * an involution of C

n

induced by a skew-hermitian form h. Assume moreover that * induces

the standard involution on C. Then any maximal order M of J can be

written M = $ nE(L) where L is an ©-lattice of V such that L = L ± L ,

L. i-modular and conversely.

l

COROLLARY 7. Let C be a division quaternion algebra over a global

field K of characteristic not 2. Let M be an order of 2 - U(C , *), n even,

n '

# induced by a skew-hermitian form. Then M is maximal if and only if

M = P n M' and M' is a maximal ^-stabl e order of C .

n

REMARKS: 3) The condition M' n $ = M is automatically satisfied if

M = # n E, E an associativ e order of #'. In that c a s e M ' C E and

M C ^ n M ' C ^ n E ^ M ,

4) Lemma 13 and Theorem 2 show that a maximal order of

tt(& ,*), n 2, K complete discrete , contains at leas t [—-—] mutually

orthogonal idempotents ([ ] the greatest integer function). For some global

fields results similar to Lemma 13 give lower bounds to the number of mutually