ARITHMETICS OF IORDAN ALGEBRAS
55
is a maximal # -stabl e order of C then M is maximal (since 01 c 0-orders
7 n
of C , M C M implies C M' C M' + 01 so M' + 01 = M' + 01 ; therefore
n'
M C J n (M1 + 01) = ^n (M' + 01) = M). To se e the converse, localize and
complete. If C splits then M maximal implies M maximal by
P P P
Corollary 1. If C is not split O l + M ' is maximal J -stabl e by (5), (6)
and (7).
q. e. d.
COROLLARY 6. Let C be a quaternion division algebra over K the
quotient field of a Dedekind ring o. Assume that every completed localiza -
tion of C splits or has no symmetric prime (in particular this is satisfied if
K is global). Let M be an order of P - U(C , K, J ), n = 3 or n even
4. Then M is maximal if and only if M = ^ n M' and M' is a maximal
J -stabl e order of C .
7 n
PROOF. If M ^ J n M ' and M' is maximal J -stabl e then M is
7
maximal by Proposition 2. To prove the converse it suffices to show it for all
/\ /\
completed localizations . If C splits this is Corollary 1. Assume C is a
division algebra. If n = 3 the lattice L of Theorem 3 must be 2i-modular
for otherwise we should have the exception (C ha s no symmetric prime by
assumption). Hence (6) or (7) cannot occur. If n is even, L is a sum of
(subnormal) planes and an even number of lines so (6) and (7) cannot appear
a s M1. Hence under the above hypothesis if M is maximal then M' is a
maximal J -stabl e order of C .
7 n
q . e . d.
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