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
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