INTRODUCTION 3 (For a nice proof see [MH, p. 103].) Recall tha t tw o symmetri c bilinea r form s ^ an d /A 2 on lattice s (i.e. , finitel y generated fre e abelia n groups ) A l an d A 2 , respectively , ar e equivalent if ther e i s an isomorphismf: A x - * A 2 suc h thatJ*/x2 « 1 . Such a form on a lattice A is of type II if ji(JC,JC) s 0 (mod2) for all x A . Otherwise, /i is of rype I. There ar e tw o fundamenta l invariant s o f a symmetri c bilinea r for m p i on a lattice A : it s rank (th e dimensio n o f A ® R) an d it s signature (= ran k - 2q, where ? is the maximal dimension of a subspace of A ® R on which /i is negative definite). It is an elementary result that the signature of a form of type II must be a multiple of 8. Indefinite unimodula r symmetri c bilinea r form s ar e completely determine d u p to equivalence by their rank and signature. The classification is as follows. (2.2) Typ e I: M s 0 ® •• e l ) e( - l ) e e ( - l ) , (2.3) Typ e II: / i a H e © H © £8 © © £ 8 , where ( l) an d ( - l ) denot e the two possible rank-1 forms, (2.4) - ( ? !)• and £ . 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 0 0 lo 0 0 0 0 0. 1 0 0 0 0 2 1 0 0 0 1 2 1 1 0 0 1 2 0 0 0 1 0 2 1 0 0 0 1 2 Note tha t th e nonzer o off-diagona l element s o f £ 8 ca n b e associate d t o th e Dynkin diagram for th e exceptiona l Li e grou p £ 8 . Not e tha t indefinitenes s force s bot h ( l ) an d ( - 1) t o appear in (2.2) and at least one H to appear in (2.3). Definite unimodular symmetric bilinear forms are a different matte r altogether. Their study forms one of the difficult, classica l fields of mathematics. To illustrate this, we presen t th e followin g astonishin g table . Le t N(r) denot e th e number o f inequivalent unimodula r type II forms which are positive definite and of rank r. (2.5) r [ N(r) 8 1 16 2 24 24 32 107 40 io $1 ! TABLE 2 (As a basic reference for the above facts, see Milnor-Husemoller [MH].)
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