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UBC Theses and Dissertations

Classes of unimodular integral symmetric positive definite matrices Norton, Peter George 1964

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CLASSES OP UNIMODULAR INTEGRAL SYMMETRIC P O S I T I V E D E F I N I T E MATRICES b y P e t e r George N o r t o n B . S c , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1963 A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M a s t e r o f A r t s i n t h e D e p a r t m e n t o f o f M a t h e m a t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A u g u s t , 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study* I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of Mathematics The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date August 31, 196k. ABSTRACT I t i s shown t h a t t h e number o f c l a s s e s o f n o n i s o m e t r i c l a t t i c e s on t h e s p a c e o f r a t i o n a l n - t u p l e s i s t h e same as t h e number o f c l a s s e s o f n x n i n t e g r a l , s y m m e t r i c , p o s i t i v e d e f i n i t e , u n i m o d u l a r m a t r i c e s u n d e r i n t e g r a l c o n g r u e n c e . A method i s g i v e n t o d e t e r m i n e t h e number o f c l a s s e s o f n o n -i s o m e t r i c l a t t i c e s ; t h i s method i s u s e d t o d e t e r m i n e t h e number o f c l a s s e s f o r n £ 16. A r e p r e s e n t a t i v e o f e a c h c l a s s o f s y m m e t r i c , i n t e g r a l , p o s i t i v e d e f i n i t e , u n i m o d u l a r 16x16 m a t r i c e s i s g i v e n . i v ACKNOWLEDGEMENT I w i s h t o t h a n k D r . R.C. Thompson f o r s u g g e s t i n g t h e t o p i c f o r t h i s t h e s i s and f o r t h e a i d he gave me i n s o l v -i n g t h e p r o b l e m . I a l s o w i s h t o t h a n k t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r f i n a n c i a l a s s i s t a n c e d u r i n g t h e p e r i o d when I was w o r k i n g on t h i s t h e s i s . i i i TABLE OF CONTENTS Page I STATEMENT OF THE PROBLEM 2 I I P-ADIC NUMBERS 3 I I I LATTICES 4 IV RULES OF CONSTRUCTION 13 V DETERMINATION OF THE CLASS NUMBER 22 h(n) f o r n £ 16 V I DISCRIMINANT MATRICES FOR LATTICES 46 OF DIMENSION 16 1 INTRODUCTION I n t h i s t h e s i s t h e f o l l o w i n g n o t a t i o n s s h a l l be u s e d : (1) Z w i l l be t h e r i n g o f r a t i o n a l i n t e g e r s and Q w i l l be t h e f i e l d o f r a t i o n a l numbers. (2) A,B,C,M,P,Q w i l l be n x n m a t r i c e s o v e r Z u n l e s s o t h e r w i s e s t a t e d . ( 3 ) The t r a n s p o s e and d e t e r m i n a n t o f P s h a l l be w r i t t e n lP~ and d e t P r e s p e c t i v e l y . P w i l l be s a i d t o be u n i m o d u l a r i f P i s a m a t r i x o f i n t e g e r s a n d d e t P = - 1. 1^ s h a l l be t h e n x n i d e n t i t y m a t r i x . (4) M a t r i x A w i l l be s a i d t o be i n t e g r a l l y c o n g r u e n t t o m a t r i x B i f t h e r e i s an i n t e g r a l m a t r i x P s u c h t h a t A = P T B P . and d e t P = 1. (5) V w i l l be t h e v e c t o r s p a c e o f r a t i o n a l n - t u p l e s . (6) A m a t r i x A i s c a l l e d s y m m e t r i c i f A = A' . I f A i s a s y m m e t r i c m a t r i x t h e n Q^i.x) w i l l be t h e f u n c t i o n on V whose v a l u e a t c o l u m n n - t u p l e x i s x T A x . (7) A s y m m e t r i c m a t r i x A w i l l be c a l l e d p o s i t i v e d e f i n i t e i f Q A i s a p o s i t i v e d e f i n i t e f u n c t i o n , ^ t ^ w i l l be t h e s e t o f n x n s y m m e t r i c p o s i t i v e d e f i n i t e i n t e g r a l m a t r i c e s w i t h d e t e r m i n a n t one. ^ w i l l be t h e s e t o f n x n sym-m e t r i c i n t e g r a l m a t r i c e s . TEXT I . STATEMENT OF THE PROBLEM L e t A and B be two i n t e g r a l l y c o n g r u e n t s y m m e t r i c i n t e g r a l m a t r i c e s . ( i . e . T h e r e i s a u n i m o d u l a r m a t r i x P w i t h A = P T B P ) . Then d e t A = d e t P T , d e t B d e t P = d e t B. I f A i s a s y m m e t r i c m a t r i x t h e n ( P T A P ) T = P T A P . Thus we se e t h a t i n t e g r a l c o n g r u e n c e i s a n e q u i v a l e n c e r e l a t i o n on jS^ w h i c h s p l i t s i n t o e q u i v a l e n c e c l a s s e s i n e a c h o f w h i c h a l l m a t r i c e s h a v e t h e same d e t e r m i n a n t . I t f o l l o w s t h a t t h e r e a r e i n f i n i t e l y many i n t e g r a l c o n g r u e n c e e q u i -v a l e n c e c l a s s e s i n ^ . I f A i s a s y m m e t r i c p o s i t i v e d e f i n i t e n x n m a t r i x and P i s a u n i m o d u l a r n x n m a t r i x t h e n P T A P i s an n x n s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i x . Thus c o n g r u e n c e i s an e q u i v a l e n c e r e l a t i o n on ^ and so s p l i t s i n t o a number o f e q u i -v a l e n c e c l a s s e s . The o b j e c t o f t h i s t h e s i s i s t o d e t e r m i n e h ( n ) t h e number o f c l a s s e s o f u n i m o d u l a r i n t e g r a l p o s i t i v e d e f i n i t e s y m m e t r i c n x n m a t r i c e s b y K n e s e r ' s method [ 2 l . h ( n ) s h a l l be c a l l e d t h e c l a s s number. I t i s w e l l known t h a t h ( n ) i s f i n i t e . ( C I ] , page 7 4 ) . 3 I I . F-ADIC NUMBERS L e t r be a r a t i o n a l number and p be a p r i m e i n t e g e r . i Then r = p \ ^  ) where m and n a r e i n t e g e r s r e l a t i v e l y -p r i m e t o p. We d e f i n e t h e p - a d i c v a l u a t i o n o f r , I r l p , t o be p"*. Then i f s i s a n o t h e r r a t i o n a l we have I r s l p = (|r| p)(J3| p) and |r+s\ p i \ r | p +|s\p. J u s t as w i t h o r d i n a r y a b s o l u t e v a l u e , y p o s e s s e s a c o m p l e t i o n w i t h r e s p e c t t o t h i s p - a d i c v a l u a t i o n . The elements o f t h i s v a l u a t i o n a r e c a l l e d P - a d i c numbers. A p - a d i c number a f o r w h i c h |a| p < 1 i s c a l l e d a p - a d i c i n t e g e r . F o r a f u l l t r e a t m e n t o f p - a d i c v a l u a t i o n s t h e r e a d e r i s r e f e r r e d t o [ 3 ] o r [ 4 ] . 4 I I I . LATTICES L e t V be t h e v e c t o r s p a c e o f r a t i o n a l n - t u p l e s . I f x j , x t , . . . , x h a r e r i n d e p e n d e n t v e c t o r s i n V , t h e n we d e f i n e t h e l a t t i c e L t o be t h e s e t o f a l l p x * p x +..,+p x where p- a r e i n t e g e r s . x . , x,,. . . , x p i s c a l l e d a b a s i s o f L and L i s s a i d t o be o f d i m e n s i o n r . I f r < n t h e n L i s s a i d t o be i n V , a n d i f r = n t h e n L i s s a i d t o be on V . We now p r o v e : Theorem ( l ) : L e t L be a l a t t i c e i n V w i t h b a s i s x . , x , . . . . , x . L e t A be a n r x r m a t r i x a n d l e t y, = £ a ; ; x ; i s r j J J T Then y, ,. . . , y r i s a b a s i s o f L i f and o n l y i f A i s a u n i -m o d u l a r i n t e g r a l m a t r i x . P r o o f : L e t x , x . , . . . x _ and y ,y_...y,. be b a s e s o f l a t t i c e L. Then v.- = T a. . x- and x. = Y b;:y;. when a - • and * j <-j J j '/ J *• J b , j a r e i n t e g e r s . T h e n ( y , ,yz »• • • >y r) = ( a i > j ) ( x l , x t . ,.xr) = U j j f e y ) i y r , - > . , y r ) . Hence ( a - t j ) ( b j j j ) = I and s i n c e d e t ( a V j ) and d e t (b- t j ) a r e i n t e g r a l we have d e t ( a t j ) = d e t (t>; j ) = - .1. C o n v e r s e l y , l e t x , , . . . , x r be a b a s i s o f L and l e t A be a u n i m o d u l a r i n t e g r a l m a t r i x . L e t i y . , y • ..y ) T = A ( x , ,xl.. . x r ) T . Th e n c l e a r l y y, ,y £,.... , y r i s a n i n d e p e n d e n t s e t o f v e c t o r s i n L. F u r t h e r , s i n c e d e t A = - 1, A"' i s an i n t e g e r m a t r i x a n d so t h e x ' s a r e i n t e g r a l l i n e a r com-b i n a t i o n s o f t h e y ' s . Hence i f z i s i n L t h e n z i s a n 5 i n t e g r a l l i n e a r c o m b i n a t i o n o f t h e x ' s and h e n c e o f t h e y ' s . I t f o l l o w s y , , y t , . . . , y r i s a b a s i s . y.E.D. From now on V w i l l be t h e s p a c e o f r a t i o n a l n - t u p l e s and (,) s h a l l be t h e i n n e r p r o d u c t on V s u c h t h a t ( e i , e j ) = where e^ = ( 0 0 . . . 0 1 0...0) w i t h 1 i n t h e i t h p o s i t i o n . L e t L be a l a t t i c e i n V w i t h b a s i s x , , x z , . . . , x r . Then t h e d i s c r i m i n a n t m a t r i x o f L i s ( ( X £ , X J ) ) and. t h e d i s c r i m i n a n t o f L, d ( L ) , i s t h e d e t e r m i n a n t o f t h e d i s -c r i m i n a n t m a t r i x o f L. ( i . e . d ( L ) = d e t ^ ( x ; , X j ) ) ) . We h a v e : Theorem ( 2 ) ; I f L i s a l a t t i c e i n V and x , , . . . x r a n d y , i » y 1 » » » » y r a r e ^ v 0 b a s e s f o r L t h e n d e t ( ( x t , x - ) ) = d e t Uy-t ,y, ) ) . P r o o f : B y t h e o r e m one we have (x , ,xt. .. x r ) = A ( y 1 y l . . . y r ) where A i s a u n i m o d u l a r i n t e g r a l r x r m a t r i x . Then U x i , x j ) ) = A( (yi ,yj ) ) A T and so d e t U x i f X j ) ) = d e t A d e t ( ( y t , y j ) ) d e t A T = d e t ( ( y 0 y j ) ) Q.E.D. Theorem ( 3 ) : I f L i s a l a t t i c e i n V and x , , x t . . . f x p i s a b a s i s o f L, t h e n l e t t h e d i s c r i m i n a n t m a t r i x o f L w i t h r e s p e c t t o x , , x s , . . . , x r be A. L e t y , , y £ , . . . y r be a n o t h e r b a s i s o f L g i v i n g d i s c r i m i n a n t m a t r i x B. Then ( a ) A i s a s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i x (b) A and B a r e i n t e g r a l l y c o n g r u e n t ( c ) I f C i s a n r x r m a t r i x i n t e g r a l l y c o n g r u e n t t o A 6 t h e n t h e r e i s a b a s i s z , . . . z r o f L g i v i n g d i s -c r i m i n a n t m a t r i x C. P r o o f : ( a ) A = ( ( x ^ , x ^ ) ) and ( x £ ,x- ) = ( x ^ , x ^ ) so A i s s y m m e t r i c . L e t ( ) = a be a r a t i o n a l r - t u p l e . , -C o n s i d e r Q ( a ) = £ a . ( x . , x ) a. = ( £ a. x- , Z a. x. ) = ( y y ) ^ 0 where y = X a.x . F u r t h e r Q (a.) = 0 i f and o n l y i f y = 0 w h i c h i s t r u e i f and o n l y i f a- = 0 f o r a l l i ( s i n c e x ' s a r e i n d e p e n d e n t ) . (b) B y t h e o r e m ( l ) t h e r e i s a n i n t e g r a l u n i m o d u l a r m a t r i x P s u c h t h a t ( x | , . . . , x ) > j r = P ( y , ,... , y r ) T . Then l( vi.»yj)) = P i ( x -t ,x^ ) ) P T and so d i s c r i m i n a n t m a t r i c e s a r e i n t e g r a l l y c o n g r u e n t . (c) L e t Q be a n i n t e g r a l u n i m o d u l a r r * r m a t r i x s u c h t h a t C = Q A Q T = y ( ( x L , x - ) ) Q T . Then i f ( z ,, z 2 ,... z r ) T = y _ l ( x , , X j , . . . , x r ) T , b y t h e o r e m ( l ) z, , z^,. . . z,, i s a b a s i s o f L ( s i n c e y i s i n t e g r a l u n i -m o d u l a r ) a nd ^ ( z ^ , Z j ) ) = C. Q.E.D. An i s o m e t r y o f t h e s p a c e V i s a l i n e a r t r a n s f o r m a t i o n o~ o f V o n t o i t s e l f s u c h t h a t ( * ^ x ) , «" ( y ) ) = ( x , y ) f o r a l l x and y i n V. L a t t i c e s L and K on V s h a l l be c a l l e d i s o m e t r i c i f K = tf L. Lemma; I f K and L a r e i s o m e t r i c l a t t i c e s i n V t h e n d ( K ) = d ( L ) and t h e d i s c r i m i n a n t m a t r i c e s o f K and L l i e i n t h e same i n t e g r a l c o n g r u e n c e c l a s s . 7 P r o o f ; L e t x , , x 2 , . . . , x r be a b a s i s o f L. L e t «" be an i s o m e t r y o f V s u c h t h a t e ( L ) = K. L e t y L - «(x-J, Then » y j . » • • • i J r i s a b a s i s o f K, and ( ( x ; , X j ) ) = ( ( <T ( x v ) , <$(x3 ) ) ) = ( ( y i , y ; ) ) . § . E . D . Prom t h e o r e m (3) and t h e above lemma we see t h a t a c l a s s o f i s o m e t r i c l a t t i c e s d e t e r m i n e s a c l a s s o f i n t e g r a l l y c o n g r u e n t m a t r i c e s . L e t L be a l a t t i c e on t h e s p a c e V o f r a t i o n a l n - t u p l e s . L e t x , , x 2 , . . . b e a b a s i s o f L. Then L i s s a i d t o be u n i m o d u l a r i f t h e d i s c r i m i n a n t m a t r i x o f L h a s i n t e g e r e l e m e n t s and d ( L ) = 1 . B y t h e o r e m (3) we s e e t h a t i f t h e d i s c r i m i n a n t m a t r i x o f L w i t h r e s p e c t t o one b a s i s i s i n t e g r a l t h e n a l l p o s s i b l e d i s c r i m i n a n t m a t r i c e s o f L a r e i n t e g r a l . B y t h e above lemma we see t h a t i f K and L a r e i s o m e t r i c l a t t i c e s and L i s u n i m o d u l a r , t h e n K i s u n i m o d -u l a r . T h i s f o l l o w i n g t h e o r e m i s known ([4] page 323-324). Theorem (4); I f S i s a n n x n i n t e g r a l u n i m o d u l a r s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i x t h e n t h e r e i s a u n i m o d u l a r l a t t i c e L on V, t h e s p a c e o f r a t i o n a l n - t u p l e s ( w i t h (,)) s u c h t h a t one d i s c r i m i n a n t m a t r i x o f L i s S. I n v i e w o f t h i s t h e o r e m , t h e o r e m (3), and t h e lemma f o l l o w i n g (3), we s e e t h a t t h e s t u d y o f c l a s s e s o f n x n i n t e g r a l s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i c e s w i t h d e t e r -m i n a n t one i s e q u i v a l e n t t o t h e s t u d y o f n o n - i s o m e t r i c c l a s s e s o f u n i m o d u l a r l a t t i c e s L on t h e s p a c e V o f r a t i o n a l 8 n - t u p l e s . H e n c e , t o c o u n t t h e c l a s s number h(n) we s h a l l c o u n t t h e number o f s u c h c l a s s e s o f l a t t i c e s . Now l e t K and L be two l a t t i c e s on V w i t h K S L . L e t x , , X j , . . . , x n be a b a s i s o f L and y , , y ,..,y^ be a b a s i s o f K. Then e a c h y i i s i n L and so i s an i n t e g r a l l i n e a r comb-i n a t i o n o f t h e x ' s . Hence: 1 t h e r e i s an n x n i n t e g e r m a t r i x A w i t h Ky, ,y z».• • ,y-n) = A ( x | ) x l , . . . x j . T hen t h e r e a r e i n t e g r a l u n i m o d u l a r m a t r i c e s P and y s u c h t h a t /•&, w i t h i ; j and 5; > o (See l l ) , page 18: S m i t h C a n o n i c a l F o r m ) . S i n c e P and Q a r e i n t e g r a l u n i m o d u l a r m a t r i c e s , i f we l e t (y,'>yi»• • • » y n ) T = p (y, . . . y t l ) T and ( x ' , . . . , x ^ ) = Q ( x , ,. . . ,x*y t h e n x ^ X j , . . . ^ , and y i > - ' - > y ^ a r e n e w b a s e s o f K and L r e s p e c t i v e l y . Then T T s i n c e ( y , ^ , ,y^) = A ( x , , x t , . . .x^) , we have (y,' >Yi » • • -y^) = PAQ(x^ , X j , . . .x^) . Hence y- = l-x{ . These iK a r e c a l l e d t h e i n v a r i a n t f a c t o r s o f K i n L. Now l e t z = p, x,' + P z x^ +•••+ p^x^, be i n L. ( i . e . p t i s a n i n t e g e r ) . Then p ; = *\" w h e r e ^ and a r e i n t e g e r s a n d 0 < K\- < . Hence Now K i s a s u b g r o u p i n t h e a b e l i a n g r o u p L and so we may c o n s i d e r L/K. The e l e m e n t z o f L goes i n t o C i l = X*\ i n L/K. 9 Then since 0 <: o*L < S i the number of elements of L/K is f i n i t e and i s in fact IT Si . Hence we have Theorem ( 5 ) : If L and K are l a t t i c e s on V with L ? K and A, , ,. .. S„ are the invariant factors of K i n L, then the order of L/K (as a group) i s TT = )det A\ . Hence there are only f i n i t e l y many l a t t i c e s M on V such that K£ M S L. Now l e t K and L be two l a t t i c e s on V. Ve define the index of K i n L, d(L , K ) to be [ L: L nK ] , the order of L / L n K (as an additive group). ¥e now prove Lemma; If K and L are l a t t i c e s on V then LAK i s a l a t t i c e on V. Proof: Let x, ,x z ,.. . ,xv be a basis of L and y, ,y a ,. .. be a basis of K. Since yt ,y t,...,y T i s a basis of V we may •A write x L = £ a — yj for 1 £ i £ r, where a ^ are rationals. Let t be an integer such that t a is inte-gral for a l l i and j . Then t x- i s an integral linear combination of the y's and so t is i n K and hence i n L r\K. Hence there are expressions l i k e c -. x. + c-,, x, + . . + c. . x • with c , , i 0 and c 1 s i n Z. which are i n L O K. Let 5 i = c l ( x, + c t l x 2 + .. + c-t-t x-t where c— are integers and c i t are the minimal positive integer such that j i i s i n L A K . 10 L e t x7 be a n y v e c t o r i n L f\K. Then x i s i n L and so x = a, x, + ... + a r x r w h e r e t h e a's a r e i n t e g e r s . L e t a r = q r c r r + r r where q r and r r a r e i n t e g e r s a n d 0^ r ^ < c ^  . C o n s i d e r X - qr jr. T h i s v e c t o r l i e s on L rvK and x^.- q r ^ = b,x, + ... + b r _ ( x r_, + r p X r where b's a r e i n t e g r a l . B u t 0& r r<. c r r and cr r i s t h e m i n i m a l p o s i t i v e i n t e g e r . Hence x - qt ^ = b,x, + b ^ x ^ + ... + b ^ x ^ , ^ 0 x r . I t e r a t i n g , we g e t x - q r $ r - q r_, £ r - i " ••• ~ <1| lt = °. Hence , % x ,... , ^  r a r e a b a s i s f o r L H K . Hence L f l K i s a l a t t i c e on V. Theorem ( 6 ) ; I f L and K a r e l a t t i c e s on V t h e n d ( K , L ) = [ L : Krih] = [ d ( L > / d ( K r , L ) J . P r o o f ; B y t h e lemma L r\K i s a s u b l a t t i c e o f L and so t h e r e a r e b a s e s x ( , X j , , . . . j X ^ o f L and y ,y ,. .. ,y^of L r»K s u c h t h a t y ; = S; x^ where St ,ll,...,Syx a r e t h e i n v a r i e n t f a c t o r s o f L n K i n L. The n ( y ; ,y- ) = U v » x j ) and so . B u t d ( L , K ) = 5.AV...S,, Q.E.D. I f K and L a r e l a t t i c e s on V w i t h d ( L , K ) = 2. Then K i s s a i d t o be adjac'.erit t o o r be a n e i g h b o u r o f L. ¥e have C o r o l l a r y ; I f K and L a r e u n i m o d u l a r m a t r i c e s on V and K. i s a d j a c e n t t o L, t h e n L i s a d j a c e n t t o K. P r o o f ; S i n c e K i s a d j a c e n t t o L we have d ( L , K ) = 2. We a l s o h ave d ( K ) = d ( L ) = 1 s i n c e K and L a r e u n i m o d u l a r . Then b y Theorem ( 6) 11 2 = d U , K ) = [ d{L)/d{LnK)]/Z = t V c U L A K ) ] 4 = [ d ( K J / d arvK)] = d ( K , L ) Q.E.D. Su p p o s e L i s a l a t t i c e on V and t h e d i s c r i m i n a n t m a t r i x o f L w i t h r e s p e c t t o a b a s i s x , , x 7 , . . . ,x^, o f L i s A. Then a s s o c i a t e d w i t h L i s a q u a d r a t i c f o r m y f t. L e t us change b a s i s i n L t o y, ,y, ,...,y*» . Then b y Theorem ( 3 ) t h e new d i s c r i m i n a n t m a t r i x B ( w i t h r e s p e c t t o y, , y 2 , . i s P T A P f o r some i n t e g r a l u n i m o d u l a r P. The new q u a d r a t i c f o r m f o r L i s y B = y (>Tap . Now a t c o l u m n v e c t o r x we have Q JTAP ( x ) = x^P^A P x = ( P x ) T A ( P x ) = Q A ( P x ) . Hence change o f b a s i s i n L r e s u l t s i n a change o f v a r i a b l e i n Q h . We s a y l a t t i c e L i s e v e n i f Q A ( x ) i s e v e n f o r c o l u m n n - t u p l e s x whose component a r e i n t e g r a l . We have Theorem \7): I f L i s a u n i m o d u l a r l a t t i c e on V t h e n L i s e v e n i f and o n l y i f ( x ^ x j s 0 (mod 2) f o r l i i < n, where x ( , X j , . . . , x w i s any b a s i s o f L. P r o o f : S u p p o s e L i s a n e v e n l a t t i c e . L e t x b e l o n g t o L. Then x = a, x, + a 1x J+...+ a m x ^ f o r i n t e g e r s a ( ,a t,. . . , a ^  . L e t a = (a, ,a2 , . . . ,a*J and l e t A = ( ( x ; , X j - ) ) . Then Q A ( a ) i s e v e n s i n c e L i s e v e n , and yfc(a) = a T A a = XL a - a - ( X | , X j ) = ( x , x ) . Hence ( x , x ) = 0 (mod 2) f o r a l l x i n L and i n p a r t -i c u l a r ( x i , x t ) 5 0 (mod 2 ) . C o n v e r s e l y , s u p p o s e ( x ^ x - J f 0 (mod 2) f o r 1 & i « n . L e t a = ( a , , a z , . . . , a ^ , ) T be an a r b i t r a r y i n t e g r a l n - t u p l e . L e t A = ( ( x ; , x ; ) ) . Then Q.(a) = a T Aa = H a*. (x,,,x. ) + °. 2 ^ . a va- (x Lx-) a n d so Q A ( a ) = 0 (mod 2 ) . Thus L i s e v e n . y.E.D. 12 C o r o l l a r y ( 1 ) : A u n i m o d u l a r l a t t i c e L on V i s e v e n i f and o n l y i f ( x , x ) S 0 (mod 2) f o r a l l x i n L. C o r o l l a r y ( 2 ) : A u n i m o d u l a r l a t t i c e L on V i s e v e n i f and o n l y i f i t h a s a d i s c r i m i n a n t m a t r i x w i t h e v e n e n t r i e s on t h e m a i n d i a g o n a l . I f a l a t t i c e ( o r a q u a d r a t i c f o r m ) i s n o t e v e n . t h e n i t i s c a l l e d odd. We s a y L = L , l L 2 where L, and L z a r e s u b l a t t i c e s o f L w i t h ( L , , L Z ) = 0 i f f o r e a c h x i n L t h e r e a r e y and z i n L , and Ltrespectively and x = y + z. Theorem ( 8 ) ; I f L i s a u n i m o d u l a r l a t t i c e on a s p a c e V o v e r t h e r a t i o n a l s y and x i s a v e c t o r i n L w i t h ( x , x ) = 1 t h e n L = Z x l J where J i s a s u b l a t t i c e o f L. P r o o f ; y« i s a s u b s p a c e o f V. L e t U be t h e o r t h o g o n a l c omplement o f Qx i n V ( i . e . V i s d i r e c t sum o f U and Q* and (Q*,U) = 0 ) . We c l a i m L = Zx 1 (L r» U ) . C l e a r l y L 2 Z x i ( L ^ U ) . L e t y be a v e c t o r i n L. Then y i s i n V so y = ax + z where a i s a r a t i o n a l a n d z i s i n U. Then ( a x , x ) = a ( x , x ) = a and ( a x , x ) = ( y - z , x ) = ( y , x ) - ( z , x ) = ( y , x ) w h i c h i s a n i n t e g e r s i n c e x and y a r e i n L. Hence a i s an i n t e g e r . And so y - ax i s i n L i . e . z i s i n L. So f a r i n a n y y i n L y = a x + z where a i s a n i n t e g e r and z i s i n L r \ U i . e . y i s i n Z x l ( U o L ) . Hence L = Z x l ( U r \ L ) . y.E.D. 13 I V RULES OF CONSTRUCTION We now g i v e t h e method o f K n e s e r 12) b y w h i c h one c a n c o u n t t h e number o f c l a s s e s o f n o n i s o m e t r i c u n i m o d u l a r l a t t i c e s on t h e s p a c e V ( r a t i o n a l n - t u p l e s w i t h (,) i n n e r p r o d u c t ) . I n t h i s s e c t i o n we s h a l l h a v e : (1) V t h e s p a c e o f r a t i o n a l n - t u p l e w i t h t h e i n n e r p r o d u c t (.,) and n \ 5. (2) I w i l l be a u n i m o d u l a r l a t t i c e on V w i t h b a s i s x ( , x t , . . . , x ^ . Then d ( l ) = 1 and ( x ; , x ^ ) i s a n i n t e g e r f o r a l l 1 5 i , j $ n. 13) I f J i s a l a t t i c e on V t h e n J ? w i l l be t h e l a t t i c e g e n e r a t e d f r o m J b y e n l a r g i n g t h e c o e f f i c i e n t d o m a i n f r o m Z t o t h e p - a d i c i n t e g e r s . We s t a t e t h e f o l l o w i n g t h e o r e m w i t h o u t p r o o f . Theorem ( 9 ) : ( [ 4 ] - T 106 . 2 ) L e t K and L be two u n i m o d -u l a r l a t t i c e s on V. Then t h e r e i s a l a t t i c e J i s o m e t r i c t o K s u c h t h a t J p = L f f o r a l l odd p r i m e s p. We s h a l l now p r o v e : Theorem ( 1 0 ) : (A) L e t I be a f i x e d u n i m o d u l a r l a t t i c e on V and l e t K be a n y o t h e r u n i m o d u l a r l a t t i c e on V. Then t h e r e i s a u n i m o d u l a r l a t t i c e L on V i s o m e t r i c t o I and a c h a i n K a = I , K , , K 2 . . . , K r r L o f u n i m o d u l a r l a t t i c e s on V s u c h t h a t d(K..,,K l) = 2 f o r 1 < i < r . I n f a c t , f o r some v e c t o r x i n ( L r > | : K . M ) - K- l M w i t h ( x , x ) i n t e g r a l 14 I f b o t h I and K a r e e v e n t h e n so a r e a l l K.. L (B) L e t J and L be two u n i m o d u l a r l a t t i c e s on V. I f J i s i s o m e t r i c t o L t h e n a n y n e i g h b o u r o f J o b t a i n e d as i n (A) i s i s o m e t r i c t o some n e i g h b o u r o f L o b t a i n e d as i n ( A ) . (C) L e t I be a u n i m o d u l a r l a t t i c e on V and l e t J be a n o t h e r u n i m o d u l a r l a t t i c e on V w h i c h i s a d j a c e n t t o I . Then f o r some x i n ( y l ) J - I , L i s i s o m e t r i c t o Z x + [ y * I | ( x , y ) i s i n t e g r a l " ^ and m o r e o v e r ( x , x ) i s i n t e g r a l . C o n -v e r s e l y , i f x i s i n (^1) - I and L =|ZX + ^ y t I j ( x , y ) i s i n t e g r a l j t h e n L i s a l a t t i c e on V w h i c h i s a d j a c e n t t o I . M o r e o v e r , i f (,x,x) i s i n t e g r a l , t h e n L i s u n i m o d u l a r . I t i s o n l y n e c e s s a r y t o t a k e a f i n i t e number o f v e c t o r s x o u t o f { j l ) - I w i t h , ( x , x ) i n t e g r a l t o c o n s t r u c t , b y t h i s m e thod, a l l u n i m o d u l a r n o n i s o m e t r i c l a t t i c e s a d j a c e n t t o I . P r o o f : L e t I and K be two u n i m o d u l a r l a t t i c e s on V. Then By t h e o r e m (9) t h e r e i s a l a t t i c e L on V i s o m e t r i c t o K s u c h t h a t L p = I p f o r a l l odd p r i m e s p. L e t x , , x 2 , . . . , x ^ be a b a s i s o f I and y , , y ^ , . . . , y n be a b a s i s o f L. Then s i n c e V has d i m e n s i o n n y. = T a- t-X, (l£ i < n) where a-v- a r e r a t i o n a l s . S i n c e If= L ? f o r a l l odd p, a -v a r e p - a d i c i n t e g e r s f o r a l l o d d p r i m e s p. Hence a\.j a r e r a t i o n a l s whose d e n o m i n -a t o r s a r e powers o f 2. Suppose 2 i s t h e l a r g e s t d e n o m i n a t o r a p p e a r i n g . Then 2*"y i s an i n t e g r a l l i n e a r c o m b i n a t i o n o f t h e x ' s f o r e a c h H i < n and so 2 r L - I . S i m i l a r l y 15 2 I s L, a n d so f o r some s u f f i c i e n t l y l a r g e r : 2 r L s . I ? 2 r L. Thus d ( I , L ) i s a power o f 2. Hence t o g e t a l l c l a s s e s o f n o n i s o m e t r i c u n i m o d u l a r l a t t i c e s on V we n e e d o n l y c o n s i d e r l a t t i c e s L s u c h t h a t d ( I , L ) i s a power o f two ( i . e . t o g e t a l l n o n i s o m e t r i c u n i m o d u l a r l a t t i c e s on V we c a n s t a r t w i t h a n a r b i t r a r y u n i m o d u l a r l a t t i c e I on V and c o n s i d e r o n l y t h o s e u n i m o d u l a r l a t t i c e s L on V w i t h d ( I , L ) a power o f 2.) S u p p o s e now I and L a r e two d i f f e r e n t u n i m o d u l a r l a t -t i c e s on V and d ( I , L ) = 2 f o r some i n t e g e r <* (<*6 1 s i n c e I ^ L ) . S i n c e I 4 L t h e r e i s t i n L - I . Then 2 ^ t i s i n L r v l f o r some m i n i m a l ^ . I f x = 2 t t h e n x i s i n ((|I)r»L) - I . So l e t x be i n ( ( ^ l ) r v L ) - L. D e f i n e K ' = ^ y f r l j ( x , y ) i s i n t e g r a l ^ . K> i s an a d d i t i v e s u b -g r o u p o f I . I f y ( and y a r e i n I - K' t h e n (2Xjy () and ( 2 x , y j ) a r e i n t e g r a l s i n c e 2x i s i n I and ( x ; y , ) and ( x , y j ) a r e n o t i n t e g e r s as y , and y z a r e n o t i n K ' . Hence 2 ( x J y , )= 1 (mod 2) and 2 ( X j y £ ) s 1 (mod 2). T h e r e f o r e ( x ^ y , - y 2 ) = (x ^ y , ) .- ( x , y 2 ) i s a n i n t e g e r a n d y, - yl i s i n K and so d ( I , K # ) = [ I : K ] = 1 o r 2. C l a i m t h a t d ( I , K ' ) = 2. I t i s s u f f i c i e n t t o show t h e r e i s z i n I and n o t i n K . N o w i s i n ((^l)r> L ) - I and so x = | ( a , x , + a 2 x 2 + . . . + a^sO where a ; a r e i n t e g e r s n o t a l l e v e n . L e t z = b,x, + ... + b ^ x ^ where t h e b t a r e i n t e g e r s . Then 16 ( 2 x , z ) = ( a , a z . . . a M ) ( ( x { , x - ) ) (b , ,b 2,... , b ^ ) T where ( ( x i , X j ) ) i s an i n t e g r a l u n i m o d u l a r p o s i t i v e d e f i n i t e m a t r i x w i t h d e t e r m i n a n t one. T h e r e a r e u n i m o d u l a r U and V s u c h t h a t U ( ( x ; , X j ) ) V = I ( t h e i d e n t i t y m a t r i x ) . So ( a [ , a z',. .. , a ^ ) (b, ,b^ ,. .. ,b^ ) T = ( 2 x , z ) where ( a / j a ^ ,.. . ,a^) U= ( a , ,&z ,... ,a^) and V(b,' ,b,',. .. , b j )' = ( b v , b 2 ,... ,b r t) . Then a t l e a s t one a, s a y a ^ , i s odd ( s i n c e one a i s o d d ) . P u t a l l b = 0 e x c e p t b j w h i c h w i l l be 1. Then ( 2 x , z ) = a; = l ( m o d 2) and so ( x , z ) i s n o t an i n t e g e r and z i s i n I - K . Thus d ( I , K ) ) = 2. S i n c e x i s i n L ( x , y ) i s an i n t e g e r f o r a l l y i n L (L i s u n i m o d u l a r ) a nd hence K n L = I r\ L. Now p u t K j = K + Zx. E a c h v e c t o r v i n K t has t h e f o r m v = j ( a , x , + a , x t + ...+ a^x^,) where t h e a ^ a r e i n t e g e r s . L e t ^ ( = H c i i x i + c i i x i +•••+ c ; ; x ; ) where c ^  i s t h e l e a s t p o s s i b l e p o s i t i v e i n t e g e r . L e t v = ^ ( a , x , +...+ a^x^) be an a r b i t r a r y v e c t o r i n K,. Then a ^ = c^^+ r n where q^and r n a r e i n t e g e r s and 0 < r ^ t c ^ ^ . C o n s i d e r v - q A ^ ^ = j ( b t x , + b l X 2+...+ b^^x^.,-*- r ^ ) where bt- a r e i n t e g r a l . B y t h e m i n i m a l i t y o f c n ^ must be z e r o . R e p e a t i n g t h i s we g e t v - q,$, - q t ^ • q« t ~ = 0 • Hence %, , f%,. .. , ^ f o r m s a b a s i s f o r K , and K, i s a l a t t i c e on V. S i n c e ( x , 2 x ) i s a n i n t e g e r and 2x l i e s i n I we have 2x i n K* and S O [ K , : K ' ] f 2. B u t x i s n o t i n K* and i s i n K, and so I K , : K'} = 2. We a l s o h a v e f l : K ] = 2 . B y t h e o r e m ( 6) we have so d (D - <Mk.) = / 17 N o t i n g t h a t I n K , = K ' , we see t h a t d ( l : K,) = [ i : I A K ] = [l : K] =2 B y g r o u p t h e o r y we have = K\(KxrM*l^ _ __kj _ Jj<l T h e r e f o r e [ i : K ] [ K ' : K « L ] = \. I : K n l T ) = 1^ : I r\ L"| . Hence [K': K n ] = d ( K , , L ) = I : I o L~\ = i d ( I , L ) . T h e r e f o r e , g i v e n I and L two u n i m o d u l a r l a t t i c e s on V w i t h d ( I , L ) =2 we have p r o d u c e d K , a l a t t i c e on V a d j a c e n t t o I w i t h d ( K , ) = 1 and d ( K , , L ) = \ d ( I , L ) = 2*"' . ¥e c l a i m t h a t K, i s u n i m o d u l a r , and i f I and L a r e e v e n K ( i s e v e n . L e t w and y be i n K ( . Then w = ax + z and y = bx + v where a and b a r e i n t e g e r s and z and v a r e i n I w i t h ( z , x ) and ( v , x ) i n t e g r a l . Then (w,y) = ( a x + z, bx + v ) = a b ( x , x ) + a ( x , v ) + b ( z , x ) + ( z , v ) . Now ( x , x ) i s i n t e g r a l s i n c e x i s i n L , ( x , v ) and ( z , x ) a r e i n t e g r a l b y c o n s t r u c t i o n and ( z , v ) i s i n t e g r a l s i n c e z and v a r e i n I . a n d so (w,y) i s i n t e g r a l . We a l r e a d y know d ( K ) = 1 and so K| i s u n i m o d u l a r . Now s u p p o s e I and L a r e e v e n . We have ( y , y ) = a ( x , x ) + 2a(x,z) + ( z , z ) . Now z i s i n I and so ( z , z ) i s e v e n , x i s i n L and so ( x , x ) i s e v e n and ( x , z ) i s i n t e g r a l b y c o n s t r u c t i o n . Hence ( y , y ) O(mod 2). Thus s t a r t i n g w i t h two f i x e d u n i m o d u l a r l a t t i c e s I and K on V we p r o d u c e L i s o m e t r i c t o K w i t h d ( I , L ) =2 , we t h e n c o n s t r u c t a u n i m o d u l a r l a t t i c e K , on V a d j a c e n t t o I w i t h d ( K ( , L ) = 2* . F u r t h e r i f I and L a r e e v e n t h e n K, i s e v e n . Now b y r e p e a t i n g t h e above argument on K , and L , and 18 so o n , we p r o d u c e a c h a i n K,,K Z,...,K P o f u n i m o d u l a r l a t -t i c e s on V s u c h t h a t L = K^, and d ( K L ,Kl_i) = 2. F u r t h e r i f I a nd L a r e e v e n t h e n e a c h K L i s a l s o e v e n . To p r o d u c e K L + 1 we t a k e x i n ( j K - n L ) - K w i t h ( x , x ) i n t e g r a l . Then K l + | = Zx + [ y e K i | ( x , y ) i s i n t e g r a l ] . Thus t o d e t e r m i n e a l l c l a s s e s o f n o n i s o m e t r i c u n i m o d -u l a r l a t t i c e s on V, we s t a r t w i t h some f i x e d u n i m o d u l a r l a t t i c e I on V and p r o d u c e a l l l a t t i c e s Zx + £ y £ I : ( x , y ) i s i n t e g r a l ] where x i s i n (£l) - I and ( x , x ) i s i n t e g r a l . We t h e n f o l l o w t h e same p r o c e d u r e w i t h e a c h o f t h e s e l a t t i c e s , a n d so on. I f we p i c k f r o m t h i s s e t a l l n o n i s o m e t r i c l a t t i c e s we w i l l have one r e p r e s e n t a t i o n f r o m e a c h n o n i s o m e t r i c c l a s s o f u n i m o d u l a r l a t t i c e s on V. Hence (A) i s p r o v e d a nd s i n c e t h e c l a s s number i s f i n i t e (C) f o l l o w s . Now supp o s e J a n d L a r e two i s o m e t r i c u n i m o d u l a r l a t t i c e s . S uppose i < j and t i s t h e i s o m e t r y t a k i n g L o n t o J . I f M i s a u n i m o d u l a r n e i g h b o u r o f L t h e n t ( M ) i s a n e i g h b o u r o f J . Hence (B) f o l l o w s . Q.E.D. An i s o m e t r y 6" o f V t a k i n g a l a t t i c e L on V o n t o L i s c a l l e d a u n i t o f L. Theorem ( 1 1 ) : (D) L e t x be i n (£l) - I and K = ( y t I | U , y ) i s i n t e g r a l * } . I f x ' = x + t where t i s i n K t h e n x / p r o -d u c e s t h e same n e i g h b o u r as x i n c o n s t r u c t i o n ( C ) . (E) I f x i s i n y l - I and 6" i s a u n i t o f I t h e n x and C ( x ) p r o d u c e t h e same n e i g h b o u r o f I i n c o n -s t r u c t i o n ( C ) . 19 (P) L e t x be i n j l - I and ( x , x ) be i n t e g r a l . L e t L = Zx + K be t h e n e i g h b o u r o f I p r o d u c e d b y x where K = ^ y € I | ( x , y ) i s i n t e g r a l ] - I f t h e r e i s a v e c t o r t s u c h t h a t x - t i s i n . I , 2 ( x , t ) H 1 (mod 2) and ( t , t ) = 1 t h e n L i s i s o m e t r i c t o I . P r o o f : (D) Ye n e e d t o show Zx + K = Zx' + [ y < I I ( x ' , y ) i s i n t e g r a l } - Now [ y € I : ( x ' , y ) i s i n t e g r a l " ] = [ y fe I : ( x + t , y ) i s i n t e g r a l ] = ( y « I : ( x , y ) i s i n t e g r a l ] = K s i n c e ( t , y ) i s i n t e g r a l . Hence i t i s s u f f i c i e n t t o show Zx + K = Z ( x + t ) + K. L e t ax + y be i n Zx + K v i . e . y i s i n K and a i s i n Z ) . Then y i s i n I and so y - a t = z i s i n I . F u r t h e r ( z x ) = ( y x ) - a ( t x ) i s a n i n t e g e r s i n c e t and y a r e i n K, and so z i s i n K. Hence a x + y = a x + a t + z = a ( x + t ) + z i s Z U + t ) + K. L e t a ( x + t ) + y be i n Z ( x + t ) + K ( i . e . y i s i n K and a i s i n Z ) . Then a t + y i s i n I , s i n c e t and y a r e i n I , and f u r t h e r , ( a t + y, x ) = a ( t , x ) + ( y , x ) i s i n t e g r a l s i n c e t a n d y a r e i n K. Hence a t + y i s i n K and a ( x - f t ) + y = ax + ( a t + y ) i s i n Zx + K. Hence Zx + K = Z ( x + t ) + K, and (D) f o l l o w s (E) I f x i s i n | I - I t h e n 6 ( x ) i s i n | 6 ( I ) -ff ( I ) = \1 - I . L e t K ( x ) = | y f I : ( x , y ) i s i n t e g r a l ] . Then 6 " ( K ( x ) ) = [s ( y ) : y € I , ( x , y ) i s i n t e g r a l ] = [ t : <S""'(t) € I , ( x , <T~'(t)) i s i n t e g r a l } = ^ t : t € <5" ' i = I and (<T ( x ) , t ) i s i n t e g r a l = K ( < r ( x ) ) . 20 Then o~ ( K ( x ) + Zx) = K( 6" ( x ) ) + Z( <T ( x ) ) a n d so & ( x ) and x p r o d u c e i s o m e t r i c l a t t i c e s . (P) L e t V,{z) = z - 2j^^j t . Then i t i s e a s y t o c h e c k t i s a n i s o m e t r y o f V. L e t w be i n Zx + K. Then w = ax + y where a i s a n i n t e g e r a n d y i s i n I w i t h ( x , y ) i n t e g r a l . C o n s i d e r X(w) = w - 2 ( w , t ) t S i n c e 2 ( x , t ) I 1 (mod 2) and x - t i s i n I we have a ( x - 2 ( x , t ) t ) i n I . Now l e t t = x + u where u i s i n V. Then s i n c e t - x i s i n I we have u i n I and so 2 t = 2x + 2u i s i n I . I n a d d i t i o n , ( y , t ) = ( y , x ) + ( y , u ) i s an i n t e g e r s i n c e y i s i n K and so y - ( y , t ) ( 2 t ) i s i n I . Hence t ( w ) i s i n I . Thus r ( L ) I . Now d ( l ) = d ( L ) = 1 and so d ( t ( L ) ) = 1. B y t h e o r e m (6) U s i n g t h e o r e m ( 1 0 ) , we c a n p r o d u c e a l l c l a s s e s o f n o n -i s o m e t r i c u n i m o d u l a r l a t t i c e s on V. We f i r s t p i c k some a r b i t r a r y u n i m o d u l a r l a t t i c e I on V and u s i n g t h e c o n s t r u c t i o n o u t l i n e d i n (C) ( h e r e a f t e r r e f e r r e d t o as c o n s t r u c t i o n ( C ) ) we f i n d a l l u n i m o d u l a r l a t t i c e s a d j a c e n t t o and n o n i s o m e t r i c t o I . We t h e n r e p e a t t h e p r o c e d u r e on a l l t h e s e n e i g h b o u r s o f I , and t h e n a l l t h e i r n e i g h b o u r s , and so on. By (B) w h e n e v e r we come t o a l a t t i c e i s o m e t r i c t o some p r e -v i o u s one we s i m p l y i g n o r e i t and a l l i t s n e i g h b o u r s . The o n l y d i f f i c u l t i e s a r e t o d e t e r m i n e when l a t t i c e s a r e i s o m e t r i c , = ax + y - 2 ( a x , t ) t - 2 ( y , t ) t = a ( x - 2 ( x , t ) t ) + y - 2 ( y , t ) t . and so I ( L ) = I and L i s i s o m e t r i c t o I . Q.E.D. 21 and t o d e t e r m i n e when we have g o t a l l c l a s s e s o f l a t t i c e s a n d c a n s t o p . Theorem ( l l ) i s o f some h e l p i n r e s o l v i n g t h e s e d i f f i c u l t i e s . 22 V DETERMINATION OF THE CLASS NUMBER h ( n ) FOR N ^ 1 6 We w i l l f o l l o w t h e method o u t l i n e d i n t h e l a s t s e c t i o n t o d e t e r m i n e h ( n ) f o r n ^ 16. We have V t h e s p a c e o f r a t i o n a l n - t u p l e s , ( n V 5 ) , w i t h i n n e r p r o d u c t (,) on V. We l e t I ^ be t h e l a t t i c e on V w i t h b a s i s e , , e t , . . . , e ^ where ( e i } e j ) = . (Then a d e t e r m i n a n t m a t r i x f o r I - n i s I ^ t h e n x n i d e n t i t y m a t r i x ) . We b e g i n b y c a l c u l a t i n g t h e n e i g h b o u r s o f I ^ b y method ( C ) . From t h i s p o i n t on a l l l a t t i c e s s h a l l be u n i m o d u l a r . ( i ) The l a t t i c e K ^ f o r m=0 (mod 4 ) . I n c o n s t r u c t i o n (C) we must t a k e a v e c t o r : x o u t o f (^1 n ) - I L e t t h i s v e c t o r be x = £ e^. Then a l l ^ a r e i n \Z and a t l e a s t one i s i n \Z - Z. B y r e l a b e l l i n g t h e b a s i s we may assume, f o r 1 £ i < m, ^ i s i n (^Z) - Z and f o r m + 1 t i ^ }i i s i n Z. L e t - y = X ^ - i + 2. \i e; where "hr a r e i t i ^ i s « r \ * i i n t e g e r s s u c h t h a t ^ ~\\. ~ ~ ? a n d "\«- - ^ (mod 2) (1 £ ii m) Then - y i s i n I ^ and ( x , y ) i s i n t e g r a l . Hence b y (D) we c a n u s e x - y = £ - ^ i n s t e a d o f x i n ( C ) . F u r t h e r t h e i s o m e t r y f d e f i n e d b y « ( e ^ ) = - e- and ( e ^ ) = e- , f o r i ^ j , i s a u n i t o f I . A p p l y i n g some i s o m e t r i e s l i k e vn t h i s t o x - y we c a n change x - y i n t o x' = ( x - y ) = j Z. e Hence b y (E) i t i s o n l y n e c e s s a r y t o c o n s i d e r x ' s l i k e \ I e- i n ( C ) . S i n c e we a r e o n l y c o n c e r n e d w i t h u n i m o d u l a r l a t t i c e s a d j a c e n t t o 1 ^ we o n l y w i s h t o c o n s i d e r v e c t o r s x o u t o f ( i l , , ) - I f o r w h i c h ( x , x ) i s i n t e g r a l . Hence we ne e d o n l y 23 c o n s i d e r v e c t o r s x l i k e \ e- where m = 0 (mod 4) i n c o n s t r u c t i o n ( C ) . We now p r o v e : Theorem ( 1 2 ) : The l a t t i c e s a d j a c e n t t o 1^ a r e K m 1 I ^ _ w where m - 0 (mod 4) and P r o o f : Prom t h e above argument we see t h a t i t i s o n l y n e c -e s s a r y t o u s e x = j <L e , where m z O (mod 4 ) , i n c o n s t r u c -t s t i o n ( C ) . B y (C) t h e a d j a c e n t l a t t i c e f o r t h i s x i s L = Zx + j y t I \ ( x , y ) i s i n t e g r a l ^ . Hence z i s i n L i f and o n l y i f z = 2_ (-ja + P ; ) e ^ + £ Vcei where p., a and | I p^ a r e i n t e g r a l . L e t z = 57 (i& + + H P , - e i H-e i n P ° r H i i m l e t $i = ^ a + p^ . Then 2 ^ = a +• 2pv- i s an i n t e g e r , 1*1 M l lc - £i = P i - P-. i s a n i n t e g e r a nd £ T. %; = \ ZT (£a + p-m ^ . i - i 2; a + \ T- p t i s i n t e g e r s i n c e mHO (mod 4 ) . Hence z i s i n K I I and L ? K l i L e t y = 7L j>i e 4 + p-e £- l i e i n K ^ H ^ . Then 2 , ^ ; - f , j P; and ^ £ S i a r e i n t e g e r s . N o t e t h a t t h i s r e q u i r e s a l l t o b e l o n g t o Z o r a l l ^ t o b e l o n g t o ( i Z ) - Z Case ( l ) : A l l ^ a r e i n t e g r a l . T hen f o r 1 < ±< m l e t p. = ^ WO 1f\ V*\ We have y = Z] ( p . + 0 , y ) e i + H p e - where £ I L p.= i £ C t i s i n t e g r a l and so y i s i n L. Case ( 2 ) : A l l a r e i n (^Z) - Z. Then f o r 1< 1* m l e t p L = £ ; - We have y = 27 (P • + D e . + 21 p ^ where a l l p ^ a r e i n Z. 24 F u r t h e r £ P; = I ( ^ ; - i) = I (. - ? S 0 (mod 2) s i n c e m= 0 (mod 4 ) . Hence y i s i n L. I n b o t h c a s e s y i s i n L a n d so 1 I _ .^S Ls K M JL 1^ „ Theorem ( 1 3 ) : ( a ) K i s i s o m e t r i c t o I ^ (b) I f mHO (mod 4) and m>4 t h e n K ^ i s n o t i s o m e t r i c t o I . try P r o o f : ( a ) K 4 i s c o n s t r u c t e d b y (C) w i t h x = \ £f( e-. L e t t = £(e , + e ^ + e s - e j . Then ( t , t ) = 1, 2 ( x , t ) = 1 and x - t = e ^ w h i c h i s i n I ^  . Hence b y ( F ) i s i s o m e t -r i c t o 1^. (b) L e t y be a v e c t o r i n K m . Then y = E %L e L where 2 f-t , ^ - (• , and \ Z. f- a r e i n t e g r a l . We c o n s i d e r 2 c a s e s . Case ( 1 ) : A l l f t a r e i n ^Z - Z. Then ( y , y ) = 27 } I l = 4 Case ( 2 ) : A l l a r e i n t e g r a l . Then ( y , y ) = L \i = J_ = 0 (mod 2 ) . Then i t f o l l o w s t h a t i n K m t h e r e a r e no v e c t o r s y w i t h ( y , y ) = 1 and so K ^ i s n o t i s o m e t r i c t o I„, . Q.E.D. C o r o l l a r y : I f m = 0 (mod 4) and m > 4 K ^ c o n t a i n s no v e c t o r s w w i t h (w,w) = 1. A b a s i s o f K m f o r m = 8, 12, 16... i s known ( 14] p. 331) and i s : y, 2 e 4 , - ( e ^ - e^, ( e H - e3 ), - ( e 5 - e^), ... , ( e ^ - em_/ ) where y = i ( e , + e t+...+ e j . Then t h e d i s c r i m i n a n t m a t r i x 25 f o r K m i s i n t h e i n t e g r a l c o n g r u e n c e c l a s s o f : 1 0 0 0 0 0 • 1 4 2 0 0 0 0 0 0 2 2 1 0 0 0 • 0 0 0 1 2 1 0 0 • 0 0 0 0 1 2 1 0 • 0 0 • 0 0 0 1 2 1 0 • • 0 • • • 0 0 0 1 2 • • 1 . 0 0 0 0 1 2 j The f o l l o w i n g t h e o r e m s c a n be f o u n d i n ( £4} , p. 326 and P. 324 j . T heorem ( 1 4 ) : I f n= 0 (mod 8 ) a n d L i s a u n i m o d u l a r l a t t i c e on V, ( d i m e n s i o n n ) , t h e n t h e r e a r e e v e n and odd l a t t i c e s a d j a c e n t t o L. Theorem ( 1 5 ) : I f "V i s t h e s p a c e o f r a t i o n a l n t u p l e s t h e n t h e r e i s an even u n i m o d u l a r l a t t i c e on V i f a n d o n l y i f n i 0 (mod 8 ). We now p r o v e : Theorem ( 1 6 ) : I f m s 0 (mod 8) K m i s e v e n . I f m=4 (mod 8) K^, i s odd. P r o o f : C o r o l l a r y ( 2 ) o f t h e o r e m ( 7) and e x a m i n a t i o n o f make t h i s i m m e d i a t e . A l t e r n a t e l y we see b y t h e o r e m ( 1 5 ) i f m = 4 (mod 8) t h e n K i s odd. I n t h e c a s e m = 0 (mod 8) b y t h e o r e m ( 1 4 ) t h e l a t t i c e I ^ h a s e v e n a n d odd n e i g h b o u r s . 26 B y t h e o r e m (12) t h e n o n i s o m e t r i c n e i g h b o u r s o f 1 ^ a r e K ^ l I m . t where t - : 0 (mod 4 ) . I f t < m t h e n K t _ L I„,_ t i s odd s i n c e i t c o n t a i n s v e c t o r s w w i t h (w,w) = 1. Hence i s t h e o n l y n e i g h b o u r o f I w,which i s n o t n e c e s s a r i l y odd and t h u s K m i s e v e n . y.E.D. ( i i ) h ( n ) f o r n ^ 7 . Lemma: I f n s 5, 6, o r 7 t h e n h ( n ) = 1 P r o o f : F o r n = 5, 6, 7 we see b y t h e o r e m ( 1 2 ) t h e o n l y l a t -t i c e a d j a c e n t t o 1 ^ i s K 4 _ L B y ( t h e o r e m ( 1 3 ( a ) ) K4. i - I f l . ^ . i s i s o m e t r i c t o 1^. Hence b y (A) and (B) t h e r e i s e x a c t l y one c l a s s on n o n i s o m e t r i c u n i m o d u l a r l a t t i c e on V and so b y t h e r e m a r k s f o l l o w i n g t h e o r e m (4) h ( n ) = 1. Q.E.D, The f o l l o w i n g t h e o r e m c a n be f o u n d i n ( C1]> p. 1 0 0 ) . Theorem ( 1 7 ) : I f a i s a p o s i t i v e number and A and B a r e p o s i t i v e d e f i n i t e H e r m i t i a n m a t r i c e s w i t h ( °* °\ ( 11 0 \ ^ o A/ i n t e g r a l l y c o n g r u e n t t o \ o 8J t h e n A i s i n t e g r a l l y c o n g r u e n t t o B. Now l e t A be a n y 4*4 i n t e g r a l p o s i t i v e d e f i n i t e s y m m e t r i c m a t r i x w i t h d e t e r m i n a n t one. C o n s i d e r \<> AV . T h i s m a t r i x i s 5*5, s y m m e t r i c , i n t e g r a l , u n i m o d u l a r , p o s i t i v e d e f i n i t e and s o , b y t h e lemma a b o v e , i s i n t e g r a l l y c o n g r u e n t t o ( 0 1 * ) Hence b y t h e o r e m ( 1 6 ) . A i s i n t e g r a l l y c o n g r u e n t t o 1^ . Thus a l l 4*4 s y m m e t r i c u n i m o d u l a r p o s i t i v e d e f i n i t e m a t r i c e s a r e i n t h e i n t e g r a l c o n g r u e n c e c l a s s o f t h e i d e n t i t y 4X4 m a t r i x a n d h ( 4 ) = 1. 27 I n a s i m i l a r way we c a n show f o r n = 3, 2 o r 1 e v e r y n n s y m m e t r i c u n i m o d u l a r p o s i t i v e d e f i n i t e m a t r i x A i s i n t e g r a l l y c o n g r u e n t t o 1^. So we have p r o v e d : Theorem ( 1 7 ) : F o r 7 t h e r e i s e x a c t l y one i n t e g r a l c o n -g r u e n c e c l a s s o f n x n i n t e g r a l s y m m e t r i c p o s i t i v e d e f i n i t e u n i m o d u l a r m a t r i c e s ( i . e . h ( n ) = 1 f o r n £ 7.) We now p r o c e e d t o d e t e r m i n e n e i g h b o u r s o f K„, where n s 0 (mod 4) and n > 4 , b y u s i n g c o n s t r u c t i o n ( C ) . ( i i i ) L a t t i c e s a d j a c e n t t o K^. We s h a l l c o n s i d e r t h e 3 p o s s i b l e c a s e s f o r t h e v e c t o r x = I e i o u t o f ( j K w ) - w h i c h w i l l be u s e d i n ( C ) . (a) I f a l l ^  a r e i n Z, t h e n x l i e s i n I m and so i n c o n s t r u c t i o n (C) we g e t I m , and no new l a t t i c e s . ( b ) Some \i a r e i n Z and some { j a r e i n (^Z) - Z. Then b y (E) we may assume t h a t £l i s h a l f i n t e g r a l f o r l i i i m and ^; i s i n t e g r a l f o r m+1 i i t n. L e t VL= Z^iec X where 19j = - \ - ^- a r e c h o s e n so t h a t ^ ^ i s i n t e g r a l f o r 1 i < m, and \ „ + ^ i s 0 o r 1, so t h a t I \ ; + X m + i + = ° (mod 2). Then y i s i n K and ( x , y ) i s i n t e g r a l a n d so b y (D) we c a n u s e x = x + y, i n s t e a d o f x , i n ( C ) . x ' has t h e f i r s t m c o o r d i n a t e s - \ and t h e (m + l ) s t c o o r d i n a t e 0 o r 1. C o n s i d e r t h e mapping t : e,;-» - eL f o r l ^ i ^ n . T h i s w i l l be a u n i t o f K ( i . e . t i s an i s o m e t r y w i t h t ( K f l ) = K w ) when t h e number o f s i g n c h a n g e s i s e v e n . Now c o n s i d e r x = 2 1 - i e; + a where a i s 0 o r 1. I f m < n, t h e n d e f i n e t ( e t ) = - e c f o r 1 i i i m a c c o r d i n g as 28 t h e c o e f f i c i e n t on e ^ i n x' i s - \ and d e f i n e t ( e ^ ) = - e„ so t h a t t p r o d u c e s a n e v e n number o f s i g n c h a n g e s . I f m = n d e f i n e t on e i as above f o r 1< i£ m - 1 and t on e^ as a b o v e . Then i n e i t h e r c a s e t i s a u n i t o f and so b y ( E ) we may u s e t ( x ' ) i n s t e a d o f x" i n ( C ) . Hence we n e e d o n l y c o n s i d e r t h e f o l l o w i n g p o s s i b i l i t i e s f o r v e c t o r x o u t o f when some c o e f f i c i e n t s a r e i n t e g r a l a n d some a r e h a l f - i n t e g r a l : = 7L X= 1 + o x l = *< ^\ — 1 z ' i s i - e J X 3 = 1 : 2 m E X * = £ + 1 2 m z - e ¥e n e e d n o t c o n s i d e r x ; as t h i s v e c t o r - i s i n K „ . x z i s e l i m i n a t e d b y (F) w i t h t = - e n . I f y = 2e„, +,, t h e n y l i e s i n K f l , and ( x ,y) i s i n t e g r a l , and x 4 = x^. + y, and so b y (D) we may e l i m i n a t e x ^. Now c o n s i d e r x , = \ Z e- . S i n c e we r e q u i r e t h a t ( x , be i n t e g r a l , we have m = 0 (mod 4 ) . Suppose m > TJ. L e t y = - j Y. e t + i L. ei; Then y i s i n K and ( x 3 ,y) i s l - I t : » n - l i n t e g r a l so b y (D) we may use x ^ = x 3 + y i n s t e a d o f x 3 i n c o n s t r u c t i o n ( C ) . We l e t t be d e f i n e d a s t ( e ^ ) = e ^ , ^ f o r 1 < i < n. Then t i s a u n i t o f K and so by (E) we may u s e t ( x ' ) = x i n s t e a d o f x' i n ( C ) . B u t t ( x ' ) = x = \ Z e $ O 3 U l where m'= 0 (mod 4) and ra'^ ^ . Hence we may assume x 3 = \ Z ei where mEO (mod 4) and m< 29 Consider x. = ? H e ; + e,^, . Since ve require (x,x) to be integral we have m = 0 (mod 4 ) . Suppose Let VW-I m y = - i £ e - + 2 - e w - i e„ + (+ \ .£ eL . Then y is in K and (x^,y) is integral. Hence by (D) we may use x' = x^+ y instead of x^. Then x' = e„, + \ e,v,+, + \ E. e^. By (E) we may use x = \ £ e -v + eh.mj., instead of x' . We have n - m < £ and n - m: 0 (mod 4 ) . Hence we may assume x_, = ^ T. e . + er with m^^ and m E O (mod 4 ) . Let the l a t t i c e produced by x i n construction (C) be called P and that produced by x be called L n, m. We now prove some lemmas about l a t t i c e s P and L n , m: Lemma: Lattice P produced by x = \ £1 ei where m E 0 (mod 4 ) and m .< ^  i s isometric to K ^ l K , . Proof: P = Zx + £yfe K | (x ,y) i s integral! . Hence y is in P i f and only i f y = 2_ ( i a + ) e t + I ^ p-e^ where a, 2p- > pt - p » i £ p- and ^  L p. are integ r a l . Let y be in P, y = £ ( i a + p. ) ec + £ pt.e4. Put TP; = i a + p- for l i i t m. Then c l e a r l y 2 and are integers. Further £ °< = £ (|a + p. ) = 2 m a + £ p. = 0 (mod 2) since m = 0 (mod 4 ) . F i n a l l y , £ p- = £ P - £ p - r 0 (mod 2). Hence y i s in K w 1 K^^and P c - K M i . K ^ . We know d(P) = 1 by construction and c l e a r l y d ( K ^ l &,>-»»,) Hence by theorem (b) ^ \ H [ o( (k>v, 1 )] [ K M I V . : p l = V di (P) J - i Thus P = K j K ^ . Q.E.D. 30 We d e f i n e t h e l e n g t h o f v e c t o r s w t o be (w,w). We s a y u and v a r e p e r p e n d i c u l a r i f ( u , v ) = 0. Lemma; I f m i | , m = 0 (mod 4) and n =0 (mod 4) t h e n K w i . K ^ ^ i s n o t i s o m e t r i c t o K^. P r o o f : We s h a l l c o n s i d e r 2 c a s e s C a s e ( 1 ) : n = 8 The o n l y p o s s i b i l i t y f o r m i n t h i s c a s e i s m = 4. When m = 4 and n = 8 we have K_, l K = K t f 1 K, and t h i s i s i s o m e t r i c t o 1^ b y t h e o r e m ( 1 3 ( a ) ) . B u t we know b y t h e o r e m (13) t h a t K f l i s n o t i s o m e t r i c t o I e a n d so K_ i s n o t i s o m e t r i c t o K„j.K . Case ( 2 ) : n > 8 F o r a r b i t r a r y * 4, «*= 0 (mod 4) K ^ c o n -t a i n s e; + e. whose l e n g t h i s 2. A l s o r e c a l l b y t h e c o r o l -l a r y o f t h e o r e m ( 1 3 ) . c o n t a i n s no u n i t v e c t o r s . L e t w be a v e c t o r o f l e n g t h 2 i n K M 1 K^-*,* Then w = u + v where u i s i n Km a n d v i s i n K^^. Hence (w,w) = 2 = ( u , u ) + ( v , v ) . S i n c e K ^ i s u n i m o d u l a r f o r a l l « , i t f o l l o w s ( u , u ) and ( v , v ) a r e n o n - n e g a t i v e i n t e g e r s , and n e i t h e r i s 1. Hence e i t h e r (u,u) = 2 and ( v , v ) = 0, o r (u,u) = 0 and ( v , v ) = 2. Thus w = u + 0 o r w = 0 + v where u i s i n K m and v i s i n K . Hence t h e v e c t o r s i n K ^ i . K whose l e n g t h s a r e 2 l i e i n two o r t h o g o n a l c l a s s e s . I n o n l y t h e v e c t o r s - ( e L + e^) w i t h 1< i < j f n have l e n g t h 2 s i n c e n >8. Suppose t h e v e c t o r s o f l e n g t h 2 i n K ca n be s p l i t i n t o 2 o r t h o g o n a l c l a s s e s . L e t U be t h e c l a s s c o n t a i n i n g e ( + e 2 . Then U c o n t a i n s e ( + e^, e ( + e^, 31 e , + + s i n c e none o f t h i s i s p e r p e n d i c u l a r t o e, + e 2 . Hence U c o n t a i n s - (e- ± ) f o r a n y i and j s i n c e n e i t h e r o f t h e s e i s p e r p e n d i c u l a r t o one o f e, + e 2 , e, + e 3 , . . . , e , + e^. Hence U c o n t a i n s a l l v e c t o r s o f l e n g t h 2 a nd t h e r e i s o n l y one o r t h o g o n a l c l a s s w h i c h i s a c o n t r a -d i c t i o n . Thus t h e v e c t o r s i n K ^ c a n n o t be s p l i t i n t o 2 o r t h o g o n a l c l a s s e s a n d so i s n o t i s o m e t r i c t o K m L K ^ . m . Lemma: I f m = 4 t h e n L^, m i s i s o m e t r i c t o K^. i * P r o o f : L,^, 4 i s p r o d u c e d b y x = \ e^ + e y . L e t t = \ ( e , + e z + e 3 - e ^ ) . Then ( t , t ) = 1, 2 ( x , t ) = 1 and x - t = e^ + e^. w h i c h l i e s i n K n . Hence b y ( P ) L^, 4 i s i s o m e t r i c t o K^. Q.E.D. Lemma: L fh, 8 i s n o t i s o m e t r i c t o K < 6 . e P r o o f : L/(, , 8 i s p r o d u c e d b y x = \ 21 e £ + e « a n d henc e x l i e s i n L ^ , 8. B u t ( x , x ) = 3 and K / 6 i s e v e n a n d so L / 6 , 8 i s n o t i s o m e t r i c t o K,6 . Q.E.D. C o r o l l a r y : L /6, 8 i s a n odd l a t t i c e . I t c a n be shown t h a t t h e v e c t o r s o f l e n g t h 2 i n L n , m f o r 8 < m < ^ , m-=:0 (mod 4) f o r m two o r t h o g o n a l c l a s s e s and t h a t t h i s i s n o t t r u e f o r K . Hence i n g e n e r a l L ^ , m i s n o t i s o m e t r i c t o K . Lemma: I f n = 0 (mod 8) t h e n L ^ , m c o n t a i n s no v e c t o r s w w i t h (w,w) = 1 when 8< m< ^ a n d m r O (mod 4 ) . P r o o f : L , m = Zx + [ y t K \ ( x , y ) i s i n t e g r a l } where 32 x = \ + e m + , , 8 £ m $ a nd m=0 (mod 4 ) . L e t v be i n L n , m. Then w = £: (p . + fr) e • + ( P m + I + a) e m ^ ( + f p e L where 2 P l , P i - p , a, \ I p. and \ £ Pi + a r e i n t e g e r s . S i n c e 2x i s i n and ( x , 2 x ) = 2 ( x , x ) i s i n t e g r a l we may assume a i s 0 o r 1. I f a = 0 t h e n w i s i n K A and so (w,w)= 0 (mod 2) and hence (w,w) = 1. Thus we may assume a = 1. T h e r e a r e two c a s e s f o r t h e p t . Case ( l ) ; A l l p • a r e i n t e g e r s . Then (w,w) = I I (p t- + \) + .i 2 1 . z. (Pm*,4- a ) + Z P,- • S i n c e p- a r e i n t e g e r s (p- + i ) >/ t and so (w,w) } -r + ( P n v,,+ a) + J- P* ^ T ^ 2 s i n c e m >,8 and so (w,w) £ 1. Case ( 2 ) : A l l p ^ a r e i n \% - Z. Then p = where t t - i s odd. We have 4(w,w) = V ( t ; + 1) + (t*»*-/+ 4) + Z t * . Now^^m>-8, so n>,16 and n - m - 2>,6. S i n c e 1 1- i s o d d , t ^ / l . Hence £ >/(n - m - 2) 1 >, 6. T h e r e f o r e 4(w,w) } E ( t ; + 1 ) * + ( t ( M « + 4) + 6 >,6. Thus (w,w) >/• | . I n a n y c a s e (w,w) ^ 1. y.E.D. C o r o l l a r y : L /6, 8 c o n t a i n s no v e c t o r s o f l e n g t h 1 and h e n c e L /£, 8 i s n o t i s o m e t r i c t o I / 6 . We now c o n s i d e r t h e t h i r d and f i n a l p o s s i b i l i t y f o r a v e c t o r x o u t o f ^K^: t ^ ( c ) A l l ^ . = - ~ where 11- i s odd and \ Z t£ = 0 (mod 2 ) . 33 T h e n x = \ Z t ; L e t t ^ = 2mL+i and l e t -r^be i « v o r i ( m ; + l ) w h i c h e v e r i s i n t e g r a l . S e l e c t i n t e g e r s °<c so t h a t + T \ i a r e e v e n f o r 1 < i £ n. L e t * be a n i n t e g e r and l e t y = Z. ( T t i e ; + * i O + x e - n - C o n s i d e r ( x , y ) = i £ t " I i s i (-n ;t; + * t „ ) + > t ^ . The n u m e r a t o r o f U , y ) i s Z (->!.•+ * i ) +X modulo 2 and so we c a n s e l e c t X e v e n so t h a t t h e n u m e r a t o r o f ( x , y ) i s c o n g r u e n t t o z e r o modulo 4. Then Z~ *'* + .2- ' ' l i + X=0 (mod 2) and so y i s i n K , and i n a d d i t i o n ( x , y j i s an i n t e g e r , a n d so b y (D) we may u s e x = x + y i n -•vi-I s t e a d o f x i n ( C ) . H e r e x ; = 21 - { e - + | t e f l. L e t * ( e ; ) = - e ^  where f o r 1 5 i i n - 1 t h e s i g n i s c h o s e n t h e same as t h a t o f t h e c o e f f i c i e n t o f e^ i n x' and t h e s i g n f o r e^ i s c h o s e n so t h a t t h e r e a r e an even number o f c h a n g e s o f s i g n . T h e n, as i n ( b ) , * i s a u n i t o f Y and so b y (E) we. may u s e ( x ' ) i n s t e a d o f x ' i n ( G ) . F u r t h e r i f y = -A e^ , f o r a s u i t a b l e X = 0 (mod 2 ) , t h e n b y (D) we may u s e <f ( x ' ) + y i n s t e a d o f «" ( x ') i n ( C ) . L e t x = ( x ' ) + y. Then x = £ ( Zl e; + t e ) where < 2 . a r e t h e o n l y v e c t o r s i s i we n e e d c o n s i d e r f o r c o n s t r u c t i o n ( C ) . Now t i s odd and so p o s s i b l e t a r e I 1, i 3, ± 5, - 7. We know 2x = f ( £ e; i s i n and so ( n - l)£ •+ \ E 0 (mod 2 ) . Hence t = 1 (mod 4 ) , and t h u s t h e o n l y p o s s i b l e t a r e - 3 , - 7 , 1, 5. To use o u r x i n (C) we must have ( x , x ) i n t e g r a l . Now ( x x) = ±JL + - I 1 v * ' 16 16. ¥e c o n s i d e r two c a s e s ( r e c a l l n ; 0 (mod 4 ) ) : 34 Case (1): n = 8 <* + 4. Then £(2o< + 13) i f t = - 7 i(2< +7) i f t = 5 (x,x) = i(2« +3) i f t = - 3 i(2o< +1) i f t = 1 But none of these i s an integer and so i f n =4 (mod 8) we get no x's i n |K n, with (x,x) i n t e g r a l , of this form. Case (2): n = 8« . Then |( * + 6) i f t = - 7 + 3) i f t = 5 (x,x) = £( <* + 1) i f t = - 3 i (<* ) i f t = 1 So i f n = 8* and <*s 1 (mod ,2) we get two x's with (x,x) integral x = 4- ( f- e • + 5e^) and x - = i ( 21 e • - 3c,,) and i f n = 8o( and cx = 0 (mod 2) we get two x's with (x,x) integral x 3 = ^ ( £ e^ • a n d x v = * We now consider 2 special cases: n = 8: Here o( = 1 and so vectors x of the above types i • •» are x = { ( t e t + 5eft) and x = i ( £ e; - 3e ). If t = x t we see (t,t) = 1, 2(x,,t) = 1 and x, - t = 2e e which l i e s i n K s. Hence by (P) we need not consider x . 8 I f 1 = | I et- , one can e a s i l y determine that i:(w) = w -2(w,z) z i s a unit of K 8 . Further, t ( x 2 ) = - e g; this vector was treated i n part (b). Hence by (E) we eliminate -\this case. So for n = 8 we get no new l a t t i c e s from x's of this type. 35 x = 16: H e r e <* = 2, and so v e c t o r s x o f t h e above is- lb t y p e s a r e * 3 = i ( £ e t - 7 e / 6 ) and x ^ = \ Z ec . U s i n g t = \ ZT e £ we c a n e l i m i n a t e x. b y ( F ) . L e t t h e l a t t i c e p r o d u c e d b y x ^ t h r o u g h (C) be J . Lemma: J i s e q u a l t o M l S ± Zx where x = j Z e. and M/S-i s a 15 d i m e n s i o n a l l a t t i c e c o n t a i n i n g no u n i t v e c t o r s . J i s an odd l a t t i c e n o t i s o m e t r i c t o K l(t , K 8 -L K g , K ,2 X 1^, L K ? J. I g o r I / 6 . P r o o f : J = Zx + fyfc K j ( x , y ) i s i n t e g r a l V . C l e a r i l y it x = ^ Z e- i s i n J and ( x , x ) = 1. Hence J i s an odd l a t t i c e . L e t w be i n J . Then w = Z (4 + ^ ) e t- where a> 2^;, £ - £• and ^ Z Ci a r e i n t e g e r s . S i n c e 2x i s i n K ^ a n d ( 2 x , x ) i s i n t e g r a l we c a n assume a i s 0 o r 1. T h e r e f o u r c a s e s : Case ( 1 ) : a = 0. Then w i s i n K ( 6 and so ( W , W ) T 0 (mod 2) and ivfr,W) ^ 1. Case ( 2 ) : a = 1. a r e i n t e g e r s n o t a l l z e r o . S uppose ^ 0. I f /• = - 1 t h e n t h e r e i s f f e 0, k 4 j , a n i n t e g e r and (w,w) = ZZ (I + f; ) >, ($ + (:) + = + ( i + | k f 2(h) - | > 1. I f ft - 1 t h e n lv,w) = £ ( i + >,<* + Jj)1 >, ( l i ) 2 = ff>1. Case ( 3 ) : a = 1. A l l J t a r e i n |Z - Z. N o t a l l ( L = - \. Then = $*'where t , = l (mod 2 ) . Hence (| +£t- ) l >, (£)* and f o r some $j (£ + £ ) > ( } ) . Hence (w,w) = £* \i +£i) > 16 ( i ) = 1. 36 Case (4): Either a = 1 and a l l %i = - j , or a = 1 a l l ^ •* are zero. In the f i r s t case v = - 1 Z ej and the second 16 i = ' w = j X et- . In both cases (w,w) = 1. Hence we can state + + ,fc that the only unit vectors i n J are - x = - \ £ e-. Thus J i s not isometric to K,t , K g l K g, L / f c , s , K a i 1^, K* J L I f f, or I / b . it F i n a l l y since x = 7-x e-k i s i n j and (x,x) = 1 by-theorem (8) we know Z = Zx JL M i r . M l f is a sublattice of J not containing - x and hence containing no unit vectors. The dimension of M ( < is 15. Q.E.D. We have now considered a l l possible vectors x out of (^Kn) - K^. We can state the following theorem. Theorem (18): (i) There are no l a t t i c e s adjacent to K^ and not isometric to K g or I 8 . ( i i ) A l l l a t t i c e s adjacent to K _ are isometric to IlZ or K 1 Z o r K 8 J-X^. ( i i i ) The only nonisometric l a t t i c e s adjacent to K J f c and not isometric to K,t or I I A are K g i . K f t , K I Z L 1^, L ( 6 , 8 , M J 5-i I, and K 8 1 I 8 . Proof; Examine (a), (b) and (c) above, noting that Ky i s isometric to 1^ (theorem (13)). Q.E.D. (iv) h(n) for n = 8. Where n = 8. The only unimodular l a t t i c e adjacent to I e and not isometric to I a i s K 8. In section ( i i i ) we have seen Kg has no neighbours that are not isometric to K 8 or ^ I 0 . Hence we have: 37 Theorem ( 1 9 ) : T h e r e a r e e x a c t l y 2 c l a s s e s o f 8X8 i n t e g r a l s y m m e t r i c u n i m o d u l a r p o s i t i v e d e f i n i t e m a t r i c e s u n d e r i n t e -g r a l c o n g r u e n c e ( i . e . h ( 8 ) = 2 ) . One o f t h e s e has a n e v e n and one h a s an odd q u a d r a t i c f o r m . ( v ) L a r g e r numbers o f v a r i a b l e s : A l a t t i c e L i s s a i d t o be d e c o m p o s i b l e i f L = J1K where J and K a r e non-empty s u b l a t t i c e s o f L. I f L i s n o t d e c o m p o s i b l e we s a y L i s i n d e c o m p o s i b l e . We s h a l l now p r o v e : Theorem ( 2 0 ) : A f i n i t e d i m e n s i o n a l u n i m o d u l a r l a t t i c e L ha s a s p l i t t i n g L = L ^ L ^ ^ ••• I"r i n t o i n d e c o m p o s i b l e s u b l a t t i c e s , w h i c h i s u n i q u e a p a r t f r o m o r d e r . P r o o f : ( a ) D e f i n e v e c t o r w i n L t o be r e d u c i b l e i f w = y + z where y and z a r e i n L, and ( y , z ) = 0, and y =£ 0, z ^ 0. w i s c a l l e d i r r e d u c i b l e i f w i s n o t r e d u c i b l e . I f w i s a r e d u c i b l e v e c t o r t h e n w = y + z where ( y , z ) = 0 and so (w,w) = ( y , y ) + ( z , z ) . S i n c e ( v , v ) > 0 f o r v ^ 0 we have (w,w) > ( y , y ) and (w,w) > ( z , z ) and (w,w), ( y , y ) and ( z , z ) a r e p o s i t i v e i n t e g e r s . Hence b y i n d u c t i o n a n y v e c t o r w c a n be w r i t t e n a s a sum o f i r r e d u c i b l e v e c t o r s . ( b ) I f u and v a r e i r r e d u c i b l e v e c t o r s i n L we s a y u-»v i f t h e r e i s a s e q u e n c e z0 ,z, , z z ,... , z t o f i r r e d u c i b l e v e c t o r s i n L w i t h z 0 = u , z t = v and ( z k , z i y , ) ^ 0. ^ i s a n e q u i v a l e n c e r e l a t i o n on t h e i r r e d u c i b l e v e c t o r s o f L, and h e n c e decomposes t h e i r r e d u c i b l e v e c t o r s o f L i n t o e q u i v a l e n c e c l a s s e s C. ,C. ... w i t h t h e f o l l o w i n g p r o p e r t i e s : 38 ( a ) ( C - ^ C j ) = 0 i f i 4 3 ( b ) I f K i i s t h e m i n i m a l s u b l a t t i c e o f L c o n t a i n i n g C; t h e n ( K t - , K j ) = 0 ( c ) T h e r e a r e o n l y f i n i t e l y many Ct-, f o r i f n o t K, , K, 1 K 2 , K , J. K a JL K r,. .. i s a n i n f i n i t e c l a s s o f s u b l a t t i c e s o f L e a c h p r o p e r l y c o n t a i n i n g t h e p r e v i o u s one. T h i s i m p l i e s L i s i n f i n i t e d i m e n s i o n a l w h i c h i s f a l s e . Hence we have K , J_ K e ... 1K^. F u r t h e r i f w i s a v e c t o r i n L t h e n w i s a sum o f i r r e d u c i b l e v e c t o r s and a l l i r r e d u c i b l e v e c t o r s a r e i n K, ,K K,...,K£. Hence w i s i n K , 1 K x J. . . -L K t . Thus L = K, 1 K 2 1 .. L K f r. B y c o n s t r u c t i o n i t i s c l e a r i s i n d e c o m p o s i b l e . Now s u p p o s e L = L, 1 L z l .. 1 L r i s a n o t h e r s p l i t t i n g o f L i n t o i n d e c o m p o s i b l e s u b l a t t i c e s . L e t w be a n i r r e d u c i b l e e l e m e n t o f L. We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t w i s i n L , and w i s i n C , . Then K, £ L, and K- ^L, f o r a n y j ^ 1* f o r t h e n L, w o u l d s p l i t . Hence we have (by r e l a b e l l i n g , i f n e c e s s a r y ) L,; 2 K-. F u r t h e r , i f L ; 4 K t-f o r some i , o r i f t = r , we have L = L , 1 L ^ i ... 1 L r ^ K, 1 K^L. ..1 = L, w h i c h i s f a l s e . Thus L ; = and t = r . Q.E.D. I n v i e w o f t h i s t h e o r e m , t o d e t e r m i n e a l l u n i m o d u l a r n o n i s o m e t r i c l a t t i c e s o f d i m e n s i o n l e s s t h a n 16, we n e e d o n l y d e t e r m i n e a l l u n i m o d u l a r n o n i s o m e t r i c l a t t i c e s o f d i m e n s i o n 16 and f i n d t h e u n i q u e s p l i t t i n g s o f t h e s e l a t t i c e s i n t o n o n d e c o m p o s i b l e l a t t i c e s . T hen a l l p o s s i b l e n o n i s o m e t r i c d i r e c t sums o f t h e s e n o n d e c o m p o s i b l e l a t t i c e s w i t h d i m e n s i o n 39 l e s s t h a n o r e q u a l t o 16 w i l l be a l l p o s s i b l e u n i m o d u l a r n o n i s o m e t r i c l a t t i c e s o f d i m e n s i o n l e s s t h a n o r e q u a l t o 16. ( v i ) E v e n 16 d i m e n s i o n a l u n i m o d u l a r l a t t i c e s Ye h a v e d e t e r m i n e d s e v e n n o n i s o m e t r i c u n i m o d u l a r l a t -t i c e s o f d i m e n s i o n 16. The r e l a t i o n o f t h e s e i s shown b e l o w ; a l i n e b e t w e e n two l a t t i c e s s i g n i f i e s t h a t t h e y a r e a d j a c e n t : Of t h e s e , o n l y K , b and K g i K f l a r e e v e n . Ye have c a l c u -l a t e d a l l l a t t i c e s a d j a c e n t t o and n o n i s o m e t r i c t o K,fc . Hence b y (A) i f t h e r e i s a n o t h e r e v e n l a t t i c e o f d i m e n s i o n 16 t h e r e w i l l be an e v e n l a t t i c e a d j a c e n t t o K 8 _ i K g . Hence we now p r o c e e d t o c a l c u l a t e t h e n e i g h b o u r s o f K e J L K f l . Ye c o n s i d e r v e c t o r s x o u t o f ( l ( K e J L K 8 ) ) - ( K g ± K f l ) . S u c h an x h a s t h e f o r m x = y + z where y and z a r e i n 2 ^ 9 * Ye f o r m L = Zx + £yfeK eJL K Q | ( x , y ) a r e i n t e g r a l ^ , and c o n s i d e r t h r e e c a s e s f o r y and z: C a s e ( 1 ) : y i s i n K g and z i s i n (^K^) - Kg. L e t v = u + w be i n K 8 1 K e . ( u i n Kg , w i n K g ) . Then L = Z ( y + z ) + [ v t K f l l K f i 1 ( v , x ) i s i n t e g r a l ] . Ye have ( v , x ) = ( u , y ) + (w,z) and s i n c e K g i s u n i m o d u l a r ( u , y ) i s i n t e g r a l f o r a l l w i n K e . Hence L = Z ( z ) + [ w e K a I (w,.z) i s i n t e g r a l " } 1 KQ . Then Z ( z ) + [ v t K 8 I (w,z) i s i n t e g r a l ] i s a d j a c e n t t o K g and s o , b y t h e o r e m ( 1 8 ) , L i s i s o m e t r i c 40 to e i t h e r K g 1 K 8 or I g 1 K g. Case ( 2 ) : z i s i n K g and y i s i n (iK e) - K 8 . Then as i n case ( l ) L i s i s o m e t r i c to K g l K g or K f t l I g . Case (3): Both y and z are i n (^Kg) - Kg. In t h i s case, L may be nonisometric to any l a t t i c e we have a l r e a d y d i s c o v e r e d . Hence we need on l y consider x = y + z where y and z are i n (^Kg) - K Q. We may use the r u l e s of c o n s t r u c t i o n (D) and (E) to a l t e r the v e c t o r s y and z. I f we f o l l o w the arguments of ( i i i ) without c o n s i d e r i n g the c o n d i t i o n (x,x) i n t e g r a l we get the f o l l o w i n g set of v e c t o r s out of 8 The remarks below w i l l a s s i s t the reader i n determining t h i s set of v e c t o r s : (1) The v e c t o r s i £ e; and ^ £ e,- + e m + l found i n (b) of ( i i i ) must have m =.0 mod 2 since twice these v e c t o r s are i n K g. Hence we get x , , X l , x 3 , x v and x y and i n a d d i t i o n 6 8 i u, = \ L e-, u 1 = \ I e; and u , = { f e^ + e 7 . By (D) with y = - K e , + e t + e 3 + e y ) + K e ^ ) - i ( e f c + e 7 ) + u ( i s e q u i v a l e n t to e f - ^ e 7 + and by (E) t h i s i s equi v a l e n t to x3. By (D) with y = -2e e, u ^ i s e q u i v a l e n t to x 6 g with y = A \ Z et-, u 3 i s e q u i v a l e n t to i ( e 7 - e f i ) , which 41 b y (E) i s e q u i v a l e n t t o x^,. 6 (2) F o r v e c t o r s o f t h e f o r m \ L t ^ e ; where t i i s 8 odd and £ t ; = 0 (mod 4) we p r o c e e d as f o l l o w s : as i n 7 t ( i i i ) ( c ) we nee d o n l y c o n s i d e r x = \ Z et- + e where t ='- 7, - 3, 1, 5. When t = - 7 t h e n b y (D) w i t h y = i 3 j{ei + e 2 ) - i ^ et- + ^ e 8 x i s e q u i v a l e n t t o | e, + ? e 2 -8 11^  e^ w h i c h b y (E) i s e q u i v a l e n t t o x, . S i m i l a r l y , when t = - 3 , 1, 5, i t may be shown t h a t x i s e q u i v a l e n t t o x„, X , , X g . ( Z ft ) Si Now c o n s i d e r t h e i s o m e t r y t : z -> z - 2 - ^ r ^ — • When J ( a , a ) a i s a v e c t o r i n K a o f l e n g t h 2, one c a n e a s i l y c h e c k t i s a u n i t o f K g . W i t h \a, e q u a l t o x 7 - x a , x 8 - x v , x / a - x^ , x 8 -x (j - X j , x / 0 - x f we f i n d t h a t t i s a u n i t o f K f l and t t a k e s (G) x j_ i n t o x ,, x^. i n t o x g , xt i n t o x / f t , x f l i n t o x J , x <j i n t o > x ^ i n t o x^_ . Hence b y (E) we nee d o n l y c o n s i d e r x , , x 3 , xs, and x 7 . U s i n g t h e s e v e c t o r s , we see t h a t f o r v e c t o r x i n j(Ka X K a ) w i t h ( x , x ) i n t e g r a l , we need c o n s i d e r o n l y t h e f o l l o w i n g : ( a ) e , + e, (b) e, + e , + ^ ( e r t + e„ + e (,+ e / 3 ) ( c ) e, + i ( e 2 + e 3 + e y + e f ) + e ? + | ( e , 0 + e „ + e, z + e, 3 16 (d) \ Z e; ( e ) e, + i ( e z + e 3 ) + l ( e , + e,„ + e„ +...+ e / 6 ) ( f ) e 4 + £ ( e z + e 3 ) + e, + i ( e / 0 + e „ ) f o r c o n s t r u c t i o n ( C ) . I n c o n s t r u c t i o n ( C ) , ( a ) w i l l g i v e IS" b a c k K ^ . B y ( G ) , (b) i s e q u i v a l e n t t o e, + f ( ^ e; - e / 6 ). 42 T h i s v e c t o r l i e s i n 2"K/6 , and so g e n e r a t e s a n e i g h b o u r o f K/j , and c a n t h e r e f o r e be i g n o r e d . I f we l e t y = - e, - es - e, - e,3 t h e n b y (D) ( c ) i s e q u i v a l e n t t o | ( e a + e 3 + e^. - es + e ,0 + e + e a - e / 3 ) and b y (E) t h i s v e c t o r i s e q u i v a l e n t t o j ( e , + e 2 + e 3 + Gq. + e y + e/ 0+ e,,+ e, £ ). B y (G) t h i s v e c t o r i s e q u i v a l e n t t o ( a ) . Thus we o m i t ( c ) f r o m o u r c o n s i d e r a t i o n . 16 I f we l e t t = 2\ez ~ e3 ) + 4"( e i ), t h e n u s i n g t h i s t i n ( F ) we may e l i m i n a t e ( e ) . I f we l e t y = - e, -e 3 - e, - e n t h e n b y (D) ( f ) c a n be r e p l a c e d b y i ^ e z . - e3 + e / o - e„ ) . By (E) and (G) t h i s v e c t o r i s e q u i -v a l e n t t o ( d ) . We have shown t h a t t h e o n l y v e c t o r x o u t o f i ( K f l A. K e) w h i c h one n e e d c o n s i d e r f o r u s e i n (C) i s (d) ( i . e . x = lb 4 Z e i )• L e t J be t h e l a t t i c e o b t a i n e d b y c o n s t r u c t i o n (C) f r o m t h i s x. Then we h a v e : (6 a Lemma: J i s Zx J - Z y l M | J f where x = \ Z e- and y = \ ~ \ Y. eL and M , ^ i s a 14 d i m e n s i o n a l l a t t i c e c o n t a i n i n g no u n i t v e c t o r s . J i s odd and i s n o t i s o m e t r i c t o K / 6 , Kfi J- K 8, 8 i . I g , M I S-1 I , L / 6 , 8 o r I,6 . P r o o f : J = Zx + ( y * K g X K g | (x,z) i s i n t e g r a l ^ . J 16 e '* o b v i o u s l y c o n t a i n s x = £ ZL e^ and y = \ Z e; - { I e; . S i n c e (x,x) = ( y , y ) = 1, two a p p l i c a t i o n s o f t h e o r e m ( 8 ) g i v e J = Zx 1 Z y 1 M | l t where M | 4 f i s a 14 d i m e n s i o n a l s u b l a t t i c e o f J . 8 ,6 L e t w e J . Then w = II (p • .+ %)eL + Z ($l+%)ei 43 6 16 where a, 2p- , 2 §. , p- - p ; , $. - ^ , £ p. , £ ZT £ i and i ZI p- + 4- X.^  £ t are i n t e g e r s . Since 2x i s i n Kg JL Kg and (2x,x) i s i n t e g r a l , we may assume a i s 0 or 1. We consider 3 cases: Case (1): a = 0. Then w i s i n K a X K g and so (w,w) = 0 (mod 2) and (w,w) ^ 1. Case (2): a = 1, w ^ X x, - y. Then (w,w) = 2 lb , S ( p - + i) + H ( f,+ ?) . Since a l l p- , f; are h a l f i n t e g r a l or i n t e g r a l we have (p^ + \) 1 Ti and ( + \) >,(Ti). ' + + r Since w = - x, - y then f o r some p • or some \i we have s t r i c t i n e q u a l i t y . Hence (w,w) = ZI (p • + j) + ^ ( + i) > 16(76 ) = 1 and so (w,w) ^ 1. Case (3): w = i x or - y, then (w,w) = 1. Hence the only u n i t v e c t o r s i n J are - x, - y. Thus J i s not i s o m e t r i c to K,^, K% ^ K^, K | 2 . l I«j > Kg X Ig, M , s l I, L , 6 , 3 or I / 6 » and M llf contains no u n i t v e c t o r s . Q.E.D. We have shown that the only l a t t i c e adjacent to K^ 1 K Qand not i s o m e t r i c to Kn,, I l f a , K ( 1 X I ¥ , L / i , 8 , K 8 JL 1Q or MIS-1 I i s the odd l a t t i c e M < t f 1 Zx -L Zy. Hence (A) gives as: Theorem (21): There are e x a c t l y 2 c l a s s e s of even unimodular l a t t i c e s of dimension 16. C o r o l l a r y : There are e x a c t l y 2 c l a s s e s of p o s i t i v e d e f i n i t e i n t e g r a l unimodular symmetric 16 16 matrices with even q u a d r a t i c form. ( v i i ) Odd 16 dimensional unimodular l a t t i c e s : 44 We have now d e t e r m i n e d t h e f o l l o w i n g e i g h t n o n i s o m e t r i c u n i m o d u l a r l a t t i c e s o f d i m e n s i o n 16: We h a v e h e r e a l l n o n i s o m e t r i c n e i g h b o u r s o f K,4and o f Kg X K 8 , t h e o n l y 2 e v e n l a t t i c e s o f d i m e n s i o n 16. R e c a l l t h e o r e m ( 1 4 ) w h i c h s t a t e s t h a t f o r n = 0 mod 8, e v e r y l a t t i c e i s a d j a c e n t t o an e v e n l a t t i c e . ( s i n c e a d j a c e n c y i s a s y m m e t r i c r e l a t i o n ) . Thus I / 6 , K , a J- 1^, K a J. I 8 , M , y i - Z x M m -L Zx -L Z y and L / 4 , 8 a r e t h e o n l y p o s s i b l e n o n i s o m e t r i c u n i m o d u l a r odd l a t t i c e s o f d i m e n s i o n 16. Hence we h a v e : Theorem ( 2 2 ) : If6 , L / 6 , 8 , M ) V i Zx L Z y , M I S L Z x , K,< , K 8 1 K 8 , K 6 JL I 8 , K g x I 8 a r e a l l t h e n o n i s o m e t r i c u n i m o d -u l a r l a t t i c e s o f d i m e n s i o n 16. ( v i i i ) h ( n ) f o r n 16 We f i r s t p r o v e : Theorem ( 2 3 ) : L , b , e , K ( < , K,z, K g , M|<(., M i s a r e nondecom-p o s i b l e . P r o o f : I f K 6 , K,t , M,* , o r M/4- s p l i t , t h e n one o f t h e f a c t o r s h a s d i m e n s i o n * 7 and h e n c e i s i s o m e t r i c t o 1 ^ f o r some n. ( t h e o r e m ( 1 7 ) ) . B u t t h i s i s i m p o s s i b l e , s i n c e K f t, K ( Z , M,^ . a nd M/s- c o n t a i n no v e c t o r s o f l e n g t h one. 45 I f L j 6 , g s p l i t , t h e n e i t h e r i t i s J ± K f o r two e i g h t d i m e n s i o n a l l a t t i c e s J and K o r i t i s L J . M f o r some l a t t i c e L o f d i m e n s i o n < 7. I n t h e f i r s t c a s e , b o t h K and J a r e i s o m e t r i c t o e i t h e r K g o r I Q ( t h e o r e m ( 1 9 ) ) , and s i n c e L ,fc,e i s odd, a t l e a s t one i s i s o m e t r i c t o I 8 . I n t h e s e c o n d c a s e , L i s i s o m e t r i c t o I f l where n < 7 ( t h e o r e m 1 7 ) . B u t L ) b , g c o n t a i n s no v e c t o r s o f l e n g t h one; c o n s e q u e n t l y , b o t h c a s e s a r e i m p o s s i b l e . F i n a l l y , i f K | 6 s p l i t s i t must s p l i t i n t o Kg i K e s i n c e K it) i s e v e n . B u t we know b y t h e lemma on page 29 t h a t Kl6 i s n o t i s o m e t r i c t o K 8 i K 8 . Q.E.D. Theorem ( 2 4 ) : The n o n d e c o m p o s i b l e n o n i s o m e t r i c u n i m o d u l a r l a t t i c e o f d i m e n s i o n 4? 16 a r e I , L i b, 8 K 8 , K l 2 , K ( 6 , M,^ and M,,. . Hence we g e t : Theorem ( 2 5 ) : The v a l u e s o f h ( n ) , t h e number o f c l a s s e s o f n»n i n t e g r a l , s y m m e t r i c , p o s i t i v e d e f i n i t e , u n i m o d u l a r mat-r i c e s u n d e r i n t e g r a l c o n g r u e n c e , f o r n i 16, a r e : n h(n«) n h ( n ) 1 1 9 2 2 1 10 2 3 1 11 2 4 1 12 3 5 1 13 3 6 1 14 4 7 1 15 5 8 2 16 8 46 V I DISCRIMINANT MATRICES FOR LATTICES OF DIMENSION 16. We a l r e a d y know a d i s c r i m i n a n t m a t r i x f o r 1 ^ and f o r K^ ( m = 0 (mod 4 ) ) . Hence we know d i s c r i m i n a n t m a t r i c e s f o r I i6 > > K 8 - L K g , K l 2 ± I<j. , Kg -L I Q. We now d e t e r m i n e d i s -c r i m i n a n t m a t r i c e s f o r L /6,g , M/5- J- Zx and M ( V 1 Zx 1 Zy. ( i ) A d i s c r i m i n a n t m a t r i x o f L /$ , e : e L ( 6, = Zx + { y * K , b I ( x , y ) € z] where x = \ £ e £ + e,. 8 Hence wfe L ) b , e , i f and o n l y i f w = T. (^a + | t )e; + ( a + L 5 c e i where 2 , , { Z S,t , | E + f , a r e i n t e g r a l i - /O J >•= I '-I and a i s 0 o r 1. Thus w = | ( b , ' e, + b ^ e ^ . +...+ b^, e^) where b's a r e i n t e g e r s . L e t X - = ^ ( a j r e , + a; 2 e 2+.. ,+&a e4- ) where ^ L i s a n y v e c t o r s u c h t h a t i s t h e l e a s t p o s s i b l e p o s i t i v e i n t e g e r . Then as i n t h e ar g u m e n t on page 16 X,, X z,.. . , X, 6 i s a b a s i s o f L / 6 , 0 . E x a m i n a t i o n o f t h e v e c t o r s i n L,6ta shows t h a t an a c c e p t -a b l e s e t o f b a s i s v e c t o r s i s : \ = 2e, X f e= e, + e h 2 l k < 8 A 1 = i ( e , + e 2 +...+e 8) + e, K = e, + e k 1 0 $ k <: 15 X,6= - i £ e i + | f ^ - i e t t + i T e • W i t h r e s p e c t t o t h i s b a s i s , t h e d i s c r i m i n a n t m a t r i x A = ( ( X„ o f L / 6 , 8 i s : A = '4 2 2 2 2 2 2 2 1 0 0 0 0 0 0 -1 2 2 1 1 1 1 1 1 1 0 0 0 0 0 0 -1 2 1 2 1 1 1 1 1 1 0 0 0 0 0 0 -1 2 1 1 2 1 1 1 1 1 0 0 0 0 0 0 0 2 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 2 1 1 1 1 2 1 1 1 0 0 0 0 0 0 0 2 1 1 1 1 1 2 1 1 0 0 0 0 0 0 0 2 1 1 1 1 1 1 2 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 3. 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 . 1 1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 2 1 1 1 :o 0 0 0 0 0 0 0 1 1 1 1 1 2 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 1 r l -1 -1 0 0 0 0 0 1 1 1 1 1 1 1 2 > ( i i ) A d i s c r i m i n a n t m a t r i x f o r M , j J . Zx: C o n s i d e r a t i o n s s u c h as t h o s e i n ( i ) l e a d us t o t h e f o l l o w i n g b a s i s f o r M ( 5 x Zx A , = 4e, X k = - e , + e k 2 1 k $ 15 it F o r t h i s b a s i s t h e d i s c r i m i n a n t m a t r i x f o r M . y l Zx i s : 16 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 -4 1 -4 :':2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -4 1 2 1 1 1 1 1 1 1 1 1 1 1 1 0 -4 l 1 2 1 1 1 1 1 1 1 1 1 1 1 0 -4 l 1 1 2 1 1 1 1 1 1 1 1 1 1 0 -4 l 1 1 1 2 1 1 1 1 1 1 1 1 1 0 -4 l 1 1 1 1 2 1 1 1 1 1 1 1 1 0 -4 l 1 1 1 1 1 2 1 1 1 1 1 1 1 0 -4 l 1 1 1 1 1 1 2 1 1 1 1 1 1 0 -4 l 1 1 1 1 1 1 1 2 1 1 1 1 1 0 -4 l 1 1 1 1 1 1 1 1 2 1 1 1 1 0 -4 1 1 1 1 1 1 1 1 1 1 2 1 1 1 0 -4 l 1 1 1 1 1 1 1 1 1 1 2 1 1 0 -4 l 1 1 1 1 1 1 1 1 1 1 1 2 1 0 -4 l 1 1 1 1 1 1 1 1 1 1 1 1 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ( i i i ) A d i s c r i m i n a n t m a t r i x f o r M | l +1 Zx 1 Zy: Ye f i n d t h a t : X,= 4e< X j = - e , + e ; 1 & j * 7 8 \, = - 2 e g + 2e^ A^ = - e, + e* 10 5 k < 15 '=7 i s a b a s i s o f M | l fL Zx 1 Zy. T h i s b a s i s l e a d s t o t h e f o l l o w i n g d i s c r i m i n a n t m a t r i x f o r M | l f l Zx -L Zy: 49 "16 -1 -1 -1 -1 -1 -1 2 0 0 0 0 0 0 0 1 ^ -1 2 1 1 1 1 1 0 0 0 0 0 0 0 0 :0 -1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 -1 1 1 2 1 1 1 0 0 0 0 0 0 0 0 .0 -1 1 1 1 2 1 1 0 0 0 0 0 0 0 0 0 -1 1 1 1 1 2 1 0 0 0 0 0 0 0 0 0 -1 1 1 1 1 1 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 8 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 2 2 1 1 1 1 1 0 0 0 0 0 0 0 0 0 2 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 2 1 1 2 1 1 1 0 0 0 0 0 0 0 0 0 2 1 1 1 2 1 1 0 0 0 0 0 0 0 0 0 2 1 1 1 1 2 1 0 0 0 0 0 0 0 0 0 2 1 1 1 1 1 2 0 > 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 J 50 BIBLIOGRAPHY 1. M. Newman, L e c t u r e Notes, U n i v e r s i t y of B r i t i s h Columbia 1957. 2. M. Kneser, K l a s s e n z a h l e n d e f i n i t e r Q u a d r a t i s c h e r Formen, A r c h i v der Mathematik, V I I I (1957) pp. 241-250. 3. B.L. van der Waerden, Modern A l g e b r a V o l . 2, ( E n g l i s h t r a n s l a t i o n ) , F r e d r i c k Ungar P u b l i s h i n g Co., New York, 1950. 4. O.T. O'Meara, I n t r o d u c t i o n to Q u a d r a t i c Forms, Academic P r e s s I n c . , New York, 1963. 

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