38 M. L. RACINE

and will be referred to as the transformation of Theorem II. 1.

COROLLARY 3. Any maximal *-stabl e order F or C is isomorphic to

ti

s I O

u

where, if F = E(L) nE(L)* a s in the preceding theorem, s is the number of

elementary divisors of H which are equal 1.

PROOF. Argue as in the proof of Lemma 11 .

Let L be a lattice of lr. A sublattice K of L is said to be a

component of L if L = K 0 K , where K = (&K) nL. An element x e L is

primitive if x I pL; L is said to be i-modular, i e S , if h(x, L) = * P for all

primitive vectors x = L; L is said to be modular if it is i-modular for some i.

The rank of L is the $ dimension of $L. Introduce the following notation:

L = L1 _ L L

?

means L = L7 0 L

?

and L. are components i = 1,2, : L ^ L',

L is isometric to L'; L ~ (a..) means the matrix of h with respect to a bas e

of L is (a,,); L ~ (a) , L is a line and the restriction of h to L ha s

matrix (a); L ~ (a.,) J_ (b ) means L = L. ± L

0

and L. ~ (a..), L . ~ (b ).

— i j ' v rs 1 2 1 -

IJ

2 — rs

Scaling by "K means multiplying h by 7\.

LEMMA 13. Any lattice L is a direct sum of modular components of

rank 1 or 2 ( i . e . L is an orthogonal sum of modular lines and planes).

0 0 . . . 0

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y © . . . ©

? © . . . © ,

s