Algebraic construction of bilinear forms over $\protect \mathbb{Z}$

THÉORIE DES NOMBRES BESANÇON

Années 1996/97-1997/98

A l g e b r a i c c o n s t r u c t i o n o f b i l i n e a r f o r m s o v e r z

M . K R Û S K E M P E R

A l g e b r a i c c o n s t r u c t i o n of b i l i n e a r f o r m s o v e r 7Z

Martin K r us kem p er

A b s t r a c t . C l i a p t e r S in [CS] gives algebraic constructions for c e r t a i n lattices. S o m e o f these c o n s t r u c t i o n s used the trace m a p . lu this n o t e we want to show t h a t by applying [Sj,[W], any bilinear f o r m over with nonzero d é t e r m i n a n t can b e c o n s t r u c t e d as a scaled t r a c e f o r m of s o m e algebra A = Z Ï [ X ] / ( f ( X ) ) where f ( X ) € 2Z[X] is monic, irreducible.

Let R b e a c o m m u t a t i v e ring with 1. À bilinear f o r m over R is a p a i r ( M , b) where M is a finitely g e n e r a t e d projective A - m o d u l e a n d b : M x M —• R is s y m m e t r i c bilinear.

If M is f r e e a n d e i , . . . , tn is a basis for M t h e n b can b e described b y t h e s y m m e t r i c n x n i n a t r i x B = ( b ( t t j ) ) over R.. Couversely any B G S y m ( n , R ) defines t h e bilinear f o r m ( Rn, B ) . T h e déterminant. <U:l{M,h) is defined as d t t ( M , b ) = d e t B . ( M , b ) is called régulai- if d t t ( M , b ) is a unit iu R. We call two forms ( M , b) a n d ( M ' , b ' ) with m a t r i c e s B , B ' G S y m ( n , R) isométrie if t h e r e exists an invertible M G M a t ( n , R ) such t h a t B = M * B ' M .

If fi : R. —• A is a r i n g m o r p h i s m such t h a t A is a finitely g e n e r a t e d p r o j e c t i v e R - m o d u l e ancl s G HoniFt(A, i?.), then t h e m a p [ x , y ) € A x A —* s ( x y ) defines a bilinear f o r m (A, s) over R. More generally, if J is an idéal in A such t h a t J is a finitely g e n e r a t e d p r o j e c t i v e /?.-module then (x, y) Ç î x l - » s ( x y ) defines a bilinear form ( J , s ) over R . W e call (Z\.s) scaled trace form of A / R .

Let f ( X ) 6 %[X] be a mouic separable polynomial of degree n (in one variable X ) . T h e n A : = Z [ X ] / ( f ( X ) ) is a free i £ - m o d u l e of r a n k n with basis 1 , X , . . . , Xn~1. We set A q A ® Q. If f is irreducible then A q = Q ( A ) dénotés t h e quotient field of A. Let T r : A q Q d é n o t é t h e trace m a p which is n o n - z e r o . E u l e r ' s l e m m a (see [L]

III—I, proposition 2, corollary) implies t h a t we have:

A * : = {c G A q | T r ( c A ) C Z } = l / ( f ' ( X ) ) A .

A n y c G A * defines a s y m m e t r i c bilinear form (A, T rc) w h e r e T rc m a p s ( a i , 02) G A x A to T r ( c a i n - i ) . Let N : A q —* Q dénoté the n o r m m a p . It is well known (compare [L] I I I - l ) t h a t if c = c u / f ' [ X ) G A^#, where c^u G A, then we h a v e d e t ( A , T r^c) = ( _ l ) « ( « - i ) / ^ V ( cu) . In particular, if c ^ 0 then d t t ( A , T rc) £ 0 a n d t h e f o r m ( A , T rc) is r e g u l a r if Co is a unit in the intégral closure. More generally, let J C A be an idéal in A a n d let c G ( T '^!)^#. Tlieu we obtain a s y m m e t r i c bilinear f o r m ( J , T r^c) if T rc m a p s ( a i , « a ) € J x î to '7V(c«i«2). Note t h a t if B is an idéal in A then B * = B ~^XA * = l / ( f ' ( X ) ) B ~^l. If c = c^u/ f ' { X ) where c„ G 1 ~² we have d e t ( l , T r^c) =

( - l ) « l » - W N { cu) N { l ) i . ( C o m p a r e [CS] page 220.)

An obvions question is the l'ollowiug: Let (MJ>) be a scaled t r a c e f o r m of A/Ziï. If ( N , b') is s y m m e t r i c bilinear such t h a t ( M , h) 0 Q is isométrie to ( N , b') <g> Q, then is

1

( N , b') also a scaled trace form of A j 221 T h e following ax a m p l e s show, t h a t t h e answer to this question is negative:

E x a m p l e s , (a) Let d € 22 be not a s q u a r e aucl set A = 22[Vd\. Let ( M , b) be a t w o - d i m e n s i o n a l s y m m e t r i c bilinear lbrni over 22. T h e n there exists s o m e c € A * such t h a t ( M , b) = ( A , T rc) if aud only if there exists a 2/-basis m i , m2 of M such t h a t t h e m a t r i x of b w i t h respect to m 1,1112 is in

Proof. Set r : = " - f f i⁶. We obtain as i n a t r i x of ( A , T r^r) with respect t o t h e 22-basis 1 , V d ,

( a ' V

\ b a d )

•

(b) W e consider t h e field extension A = Q [ \ / d ] / Q . It is easy t o chek t h a t a two- d i m e n s i o n a l s y m m e t r i c bilinear f o r m ( M , b) over Q is scaled t r a c e f o r m of A / Q if a n d only if 0 ^ —det(M, b) is a n o r m .

(c) Let A = ^ ^ T j . Since A

is a principal idéal d o m a i n , e x a m p l e (a) describes ail scaled t r a c e f o r m s of A j A t w o - d i m e n s i o n a l f o r m of d é t e r m i n a n t —4 is a scaled t r a c e f o r m of A / 2 2 if and only if it is given by a m a t r i x of t h e followig t y p e

{ ^ J |« = ± 2 , b = 0 or a = 0 ,b = ± 2 } .

T h e s e m a t r i c e s describe only two différent i s o m e t r y classes of f o r m s over 22. It is well known t h a t t h e r e exists more thau two différent isometry classes of s y m m e t r i c bilinear forms over 22 with d é t e r m i n a n t —4 (See [CS] page 3(52.); m o r e precisely, t h e f o r m given by t h e m a t r i x

U

is not a scaled t r a c e torni ot Aj22. Siuce over Q t h e r e exists only one i s o m e t r y class of forms w i t h d é t e r m i n a n t —4 we see t h a t t h e answer to t h e above question is negative.

T h e following p r o b l e m remains opeu: Let d 6 22 b e not a square a n d

A = Z[y/2j.

T h e n d e t e r m i n e ail isometry classes of scaled t r a c e forms of A / Z . Let D 6 M a t ( n , 22).

Let X d ( X ) = d e t ( X Eⁿ - D ) e 22[X] be t h e characteristic polynomial of D . T h e next l e m m a was shown in [T]. Other p roofs and généralisations of this resuit can b e found in [CP], [IS], [W].

L e m m a 1. (Taussky) Let B € S y 111.(11,, 22) such. that det. B ^ 0. Suppose there exists some M € M a t ( n , Z ) such. Unit B M ' = M B a n d \ . \ i ( X ) is irreducible. Let A : =

2Z[X}/(xm{X)). Then U k i t txisls su nu <• G A q und /j ^t, . . . , hⁿ G A such that Zfby + . . . + 2Zl)ⁿ is un idéal in A and B = ( T r ( c b^tb j ) ) . Furtherrnore ( 6 1 , . . . , bⁿ) G Q { A )ⁿ is an eiytnvector of M G M a t ( u , Q { A ) ) .

N o t e t h a t t h e above c is in (2£l>i + . . . 4- 2Sbn)*.~ Given s o m e B G S y m ( n , 2Z) we can always fincl s o m e M such t h a t B M1 — M B : Choose any S G S y m ( n , ZS) and set M : = B S . T h e n B M * = B { S B ) = M B ( C o m p a r e ([CP]).

L e m m a 2. F o r a n y B G S y m ( n , Z ) , det B ^ 0 there exists M G M a t ( n , Z ) such that B M4 = M B a n d X m ( ^ ) Z [ X ] is irrcduvible. F u r t h e r r n o r e we m a y assume that X m { X ) is totully real, that is ail zéros of x m ( X ) are real.

Proof. Let N = ( X j j ) b e t h e s y m m e U i c u x n m a t r i x where t h e coefficients X i j = X j i are new i n d e t e r m i n a t e s . Choose C G G L ( n , Q) such t h a t C^lB C is a diagonal m a t r i x . Since we m a y view C ~lN { C ~1) ' as a. s y m m e t r i c m a t r i x with i n d e p e n d e n t indetermi- n a t e s as coefficients, by [S] or [W] tlu- chanicteristic p o l y n o m i a l of ( C^tB C ) C ~¹N ( C ~^l)^t is irreducible a n d it is also t h e e h u r a c t r i i s l i e p o l y n o m i a l of B N . By H i l b e r t ' s irre- ducibility t h e o r e m , t h e r e exist «t ï = a G Q such t h a t x/s ( ai j) ( ^ ) is irreducible. If we choose a G ZSsuch t h a t ail <i<i^tJ G ZS, then \B(ua^{t J}){X) = aⁿXB(a,^J)(<¹~¹X) is irreducible.

Hence we set M : = B ( u a{ J) . We have B M ' = £ ( « « , , ) ' £ < = M B .

By [S], we m a y choose t h e above, such t h a t Xfî(a,^J)(^) is totally real. B u t t h e n XB(ua^{i }})\X) is t o t a l l y real as well. •

Using t h e above n o t a t i o n s we have shown:

T h e o r e m . Let ( M , b ) be a biliutur f o r m over ZS such that d e t ( M , b ) 0. Then there exist a monte, ir rtdacible / ( X ) G Z [ X ) , some idtal 1 C A : = 2 S [ X } / ( f ( X ) ) , and c G {Z2)* such that ( M , b) = ( J , T rc) . W<-: m a y a s s u m e that f { X ) is totally real.

R e m a r k s , (a) T h e resuit holds m o r e generally for ( M , b) over R , d e t ( M , b) ^ 0 where R is t h e intégral x l o s u r e of in some finite field extension F / Q and M is finitely g e n e r a t e d free. W e can also choose f l = k[X] w h e r e k is a field.

(b) Let B G S y m ( n , Z ) such t h a t det B 0 a n d t h e r e exists s o m e c G A q and b i , . . . , bn G A such t h a t B = (Tr(eb;bj)}. Let B ' G S y m ( n , ZS) such t h a t t h e r e exists C G M a t ( n , Z ) with d e t . 0 = ± 1 and B1 = 0 * 8 0 . T h e n t h e r e exists some d G A q a n d « ! , . . . , « „ G A such t h a t B ' = (TV(c'm,-«,-)). F u r t h e r r n o r e , for t h e ideals we have ZSbi-(-... + ZSbn = Z u i + . . . -f 7Zan.

Proof. By [CP] III 5.2, t h e r e exists M G M a t { n , Z ) such t h a t B M * = M B and /I = Z [ X ] / ( x m ( X ) ) . T h e n

(6, tyW(6, É)-1)(C, <56') = C ' B M ' O = ( Ct5 C ) ( CtA / ( C<) "1)<.

Q u e s t i o n . Let ( M , l>) be a bilim-ar form over R, where R. is the intégral closure oi 2Z 'm s o m e finite fiel cl extension F / Q , such tliat M is uot; a free but a p r o j e c t i v e /2-module.

C a n we realize ( M , b) as scaled trace l'onn of s o m e /?-algebra A?

R E F E R E N C E S

[CP] C o n n e r , P. E. , Perlis, R. , A survey of trace f o n n s of algebraic n u m b e r fields.

World Scientific, Singapore, 1984.

[CS] Conway, J . H. , Sloane, N. J . A . Spliere packings, lattices a n d groups. Berlin, Heidelberg, New York, Springer 1988.

[IS] Ischebeck,' F. , Scha,rlau, W. , H e n n i t esche uncl o r t h o g o n a l e O p e r a t o r e n ûber k o m m u t a t i v e n Ringen. M a t h . Ami. 200, 327-334 (1970).

[L] Lang, S. , Algebraic n u m b e r tlieory. fteading, M a s s a c h u s e t t s , Addison Wesley 1970.

[S] Scharlau, W . , On trace f o n n s of algebraic n u m b e r fields. M a t h . Z. 196, 125-127 (1987).

[T] Taussky, O. , On t h e shnilarity t r a n s f o r m a t i o n between an intégral m a t r i x with irreducible characteristic polyuomial and its transpose. M a t h . A n n . 166, 60-63 (1966).

[W] W a t e r h o u s e , W . C. , Scaled trace f o n n s over n u m b e r fields. Arch. M a t h . 47, 229-231 (1986).

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