28

M. L. RACINE

The proof of Lemma 6 yields that the number of isomorphism c l a s s e s of maxi -

mal orders of K is the number of equivalence c l a s s e s of the following

n

identification on the group Q/„n where Q is the clas s group: g ~ g

Q

Collecting results we have the following proposition.

PROPOSITION 3. Let C be the split field over K, £ = M(C ) s K .

' ° n ' n

Maximal orders of 9 coincide with maximal orders of K and the

n

isomorphism c l a s s e s of maximal orders of ? are in one to one correspondence

with equivalence c l a s s e s of the group Q/Q under the further identification

-1

g ~ g .

REMARK 1. If | Q/Q | is finite the number of isomorphism c l a s s e s of

maximal orders of # is the number of elements of Q/Q satisfying x = 1

1

plus 1/2 the number of elements satisfying x ^ 1.

Let C = K , the split quaternions. We need the following lemma.

LEMMA 7 ([44], p. 35). Let V be a vector spac e of dimension 2n

over K and g(x, y) a non-degenerat e alternating form on V. Let L be an

o-lattice of lr. Then there exist s a bas e {x, , . . . , x , y , , . . . , y } of V/K

1 n 1 n

and (fractional) o-ideals a , . . . , a such that g(x., x.) = g(y., y.) = 0,

g(x,,y. ) = 5 . . (in other words a symplectic base ) and

L = ox. 0 Q.y. 0 ox

0

0 Q_y. 0 . . . 0 ox 0 a y ; a, D ci D . . . D a and

i l l L L C . n n n l ^ , n

thes e ideals are uniquely determined by L and g.

Let p=U(C , J ) =U(C) ^U(K.J) where 0 = ( ? \), that is ,

n -y n ^dn O \-l 0/

the symplectic involution. Maximal orders of ^ are of the form E(L) n ?,

L an o-lattice of y a 2n-dimensional K-vectorspace. To determine which