Appendix 1. Quadratic Forms
The theory of symmetrie bi linear forms is intimately related to the theory of quadratie forms. In fact, over a ring in whieh 2 is a unit, the two theories are indistinguishable. For this reason, it seems advisable to give abrief deseription of quadratie forms, even though our emphasis is on bilinear forms. Let X be an R-module. As usual we assurne that the ring R is eommutative with I.
Definition. A quadratie form on X is a funetion q: X --> R sueh that
q(:n:)=:x 2 q(x)
für all 'XE R, and such that the funetion (xIY) on X x X defined by (xly)=q(x+ y)-q(x)-q(y) is bilinear over R. As an example, if ß is abilinear form (not neeessarily symmetrie), then the funetion q(x)=ß(x,x) is clearly quadratie, with assoeiated symmetrie bi linear form (xIY)= ß(x,y)+ ß(y, x). If X is projeetive, then every quadratie form on X ean be obtained in this way from a (non-symmetrie) bi linear form. Two bilinear forms ß and ß' give rise to the same quadratic form if and only if the difference ß- ß' is sympleetie. Note the identity (xlx)=2q(x). Thus the symmetrie bilinear form (xIY) assoeiated with a quadratie form must satisfy the eongruenee (xlx)=O mod 2R for all x. If 2 is not a zero-divisor in R, then any symmetrie bi linear form satisfying this eongruenee is assoeiated with a unique quadratie form q(x)=t(xlx). In partieu/ar, if 2 is a unit then every symmetrie bilinear form (xIY) comes from a 11l1ique quadratie form t(xlx). Appendix I. Quadratic Forms 111
Definition. Let (Xl' ql)' ... , (Xn, qn) be modules with quadratic forms over any commutative ring R. The orthogonal sum Xl (j) ... (j) Xn is defined to be the direct sum of the modules Xi with quadratic form q defined by the equation " q(xI(j)···(j)xn)= L.,qi(Xi) summed over 1 ~ i ~ n.
The associated bi linear form (x!y) on X = Xl (j) ... (j) X n is the ortho• gonal sum of the associated bilinear forms (X!Y)i on the Xi' Hence (x!y) is an inner product if and only if each (X!Y)i is an inner product. Definition. We will call the pair (X, q) a quadratie inner produet space, if X is finitely generated projective, and if the bilinear form (x!y) asso• ciated with the quadratic form q is an inner product on X (Chapter I, § 1.1). We observed in Chapter I, § 4 that the Witt Cancellation Theorem (4.4) is not true for inner product spaces in characteristic 2. It is interesting to note that the analogue of (4.4) for quadratic inner product spaces is true in any characteristic. That is, if
(Xl' ql)(j)(X, q)~(Xz' qz)(j) (X, q), where (XI,ql)' (Xz,qz), and (X,q) are quadratic inner product spaces over a field, then (Xl' ql)~(XZ' qz). This is proved in [Chevalley, p.16] or [Bourbaki, p. 71]. The following basic remark is due to [Fröhlich, 1969] and in• dependently to [Sah]. If Xl is asymmetrie bilinear form module and Xz is a qUo1dratic form module, then the tensor product Xl ® X z is a quadratic form module. In fact, given a symmetric bi linear form ßI on Xl and a quadratic form qz on Xz, there is a unique quadratic form q on XI®XZ satisfying the equations, both being necessary for the definition, q (Xl ®xz) = ßI (Xl' Xl) qz (X z) and (Xl ®XZ!YI ® Yz) = ßI (Xl' Ytl (xz!Yz)· Note the isomorphism (1)®Xz ~ Xz' Ifboth Xl and Xz are quadratic form modules, then using the associated bi linear form (XtlYI) as ßI it follows that XI®XZ is also a quadratic form module. The quadratic form q on XI®XZ is determined by the equations q (Xl ® Xz) = 2ql (Xl) qz (X z) and (Xl ® XZ!YI ® Yz) = (XI!YI) (X z!Y2)· The factor of 2 is surprising but necessary. This factor was incorrect1y left out in [Bourbaki, v. 24, p. 137]. 112 Appendix I. Quadratic Forms
The Witt algebra WQ (R) of quadratie inner produet spaees over R ean now be defined as folIows. (Compare [Bass], [Sah].) A quadratie inner produet spaee (X, q) is said to be splir if the module X eontains a direet summand N with N=Nl. and q(N)=O. Two quadratic inner produet spaees (Xl' ql) and (X2, q2) belong to the same Witt dass if
(Xl' ql)ffi(SI' q~)~(X2' q2)ffi(S2' q~) where the (Si' q;) are split. The Witt dasses of quadratie inner produet spaees over R now form the required algebra WQ(R). Clearly WQ(R) is a eommutative assoeiative algebra over the Witt ring W(R). This algebra does not possess a 1 element in general. There is a eanonieal augmen• tation a: WQ(R)- W(R), and the produet of two arbitrary elements Wl and w2 in WQ(R) is deter• mined by the identity wl w2 = a(wl ) w2 • Of course if 2 is a unit in R, then a is an isomorphism. In the ease of a field F, it is easily verified that every quadratie inner produet spaee over F is the orthogonal sum of a split quadratie inner produet spaee and an anisotropie quadratie inner produet spaee (i. e., one with Q (x)::j= 0 for x::j= 0). Furthermore, any two split spaees of the same rank are isomorphie. Using the Witt eaneellation theorem, it follows that every Witt dass in WQ(F) possesses one and up to isomorphism only one anisotropie representative. Suppose in particular that F is a field of eharaeteristie 2. Then the bi linear form (xIY) assoeiated with a quadratie form is neeessarily sym• pleetie (xlx)=2q(x)=0. Thtrefore the augmentation homomorphism WQ(F)- W(F) is identically zero. For if (X, q) is a quadratie inner produet spaee, then X possesses a sympleetie basis (Chapter I, § 3.5), and henee represents the zero element of the Witt ring W(F). An important invariant of a quadratic inner produet spaee in ehar• aeteristie 2 has been defined by C. Arf. Let p: F-F denote the additive homomorphism fJ (~) = ~2 +~. Choosing a sym• pleetic basis Xl"'" x n ' Yl"'" Yn for X, with inner produet matrix G ~), Arf showed that the residue dass of q(xl ) q(Yl)+ ... +q(xn) q(}'n) Appendix 1. Quadratic Forms 113 modulo fJ (F) is an invariant of (X, q). This Arf invariant depends only on the Witt class of (X, q). and hence gives rise to an additi\e surjection LI: WQ(F)->F/pF. The kernel of LI has been computed by H. Sah as fallows. Let 1 c W(F) be the fundamental ideal. Then kerncl(LI)= I· WQ(F). The resulting additive isomorphism
WQ(F)/I· WQ(F)~F/.pF should perhaps be regarded as an analogue of pfister's isomorphism
I/I 2 ~ F· / F· 1 . (Compare Chapter III, § 5.2.) Here is a simple example to illustrate the Arf invariant. Let (X, q) be a quadratic inner product space of rank 2. We continue to assume that the field F has characteristic 2. Ta any basis x, y with (xIY)= 1 we associate the residue class q(x) q(y) (mod gafF)). This is an invariant of (X, q). for under an elementary change of basis .x = x + er:y we have q (.x) = q (x) + er: + er: l q (y), and therefore q(x) q(y)=q(x) q(y)+ f,J(etq(y)). From this formula we see that the quadratic form q represellts 0 if ami only if q(x) q(y)=O mod p(F). For if q(x) q(y)=O and q(Y)=FO, then we can choose 'l. so that q (x) q (y) = O. In fact we can actually choose a symplectic basis x, y with q(x)=q(y)=O, simply by setting y= ßx+ y and choosing ß appropriately. IJ the field F is perfect, then the residue class q(x) q(y) mod gJ(F) is always a complete invariant Jor the quadratic inner product space. Proo! We may assume that q(x) =1=0. Choosing an arbitrary re• presentative Ja =q(x) q(y)+ fJ(er:) for the Arf invariant. the symplectic basis ~ x=x/Vq(x) y=:xx+.n/q('() will satisfy q(x)= 1, qCY)=Ll a. Thus Ll a determines the isomorphism class of (X, q). 0 In particular, if F is a finite fielel oJ characteristic 2 it follows that there are precisely two isomorphism classes of qlladratic inner product spaces oJ rank t\\'o over F. For by inspecting the additi\'e exact se• quence 0 -> F2 -> F ~ F we see that the cokernel Fi p (Fl has order two. Appendix 2. Hermitian Forms
Let R be an associative ring with 1, not necessarily commutative. Byan involution of R (or more precisely an "involutory anti-automorphism") is meant an additive homomorphism a H aJ from R to itself satisfying (aß)J = ßJ aJ and (aJ)J = a
for all CI. and ß. Note that 1J = 1. Examples. If R is commutative, then the identity map of R is an involution. For any multiplicative group n, the integral group ring zn possesses a canonical involution which maps each group element (J to (J-l. (Compare [Wall, § 5] as weil as [Gel'fand-Mishchenko].) The ring of n x n matriees over a commutative ring has a eanonical involution which maps eaeh matrix to its trans pose. Let R be any fixed ring with involution, and let X be a left R-module. Definition. A hermitian form on X is a function cp: XxX-4R, whieh is R-linear in the first variable and satisfies cp (y, x) = cp (x, y)J. It follows that cp(x, y) is bilinear over Z, and that cp(ax, ßy)=a cp(x, y) ßJ. If the correspondence YHCP( ,y) from X to HomR(X, R) is bijective, then cp is called a hermitian inner product, and the pair (X, cp) is called a hermitian inner product module. lust as for symmetrie inner produet spaees and quadratic inner product spaees, one can define the coneept of a split hermitian inner product space. Working modulo these split spaees, we obtain a Witt group of hermitian inner produet spaces over R. The notation W(R, J) will be used. In the eommutative case, W(R, J) has a natural ring Appendix 2. Hermitian Forms 115 structure. If J is the identity involution, then this coincides with the ordinary Witt ring W(R).
If X is a free R-module with basis e1 , ••. , en , then evidently any hermitian form rp on X is completely characterized by the matrix [rp(ei, ek)], which is subject only to the requirement that rp (ek, ei) = rp (ei' ek)J· The form rp is actually a hermitian inner product if and only if this matrix [rp(e i, ek)] is invertible. Now suppose that the ring R is commutative. Then the set
Ro = {aERlaJ =a} of fixed points forms a subring of R. The determinant of the matrix [rp(e i, ek)] is evidently an element of Ro. If we choose some new basis e~, ... , e~ for X, the determinant will be multiplied by some arbitrary element in the image of the homomorphism
norm: Re-.R~ defined by norm(a)=aaJ. Definition. The (multiplieative) residue cIass of det [rp (ei' e)] modulo norm (Re) is called the determillant of the hermitian spaee X. This determinant is a surprisingly powerful invariant. Compare Examples 2 and 4 below. In the eommutative ease, every hermitian form over R gives rise to a quadratie form Q(x)=rp(x, x) over Ro' In particular, this funetion Q takes values in Ru and the asso• ciated form (xIY) = Q(x+ y) -Q (x)- Q(y) =rp(x,y)+rp(y,x) is bilinear over Ru. Suppose now that F is a field with non-trivial involution. It follows from Galois theory that F is a quadratic Galois extension of the fixed field Fu. Jacobson theorem. Two hermitian inner product spaces over F are isomorphie if alld only if their underlying quadratie spaees are isomorphie over Fo . In other words X is isomorphie to Y as hermitian spaee over F whenever there is an Fo-linear mapping from X onto Y preserving the quadratie funetion Q (x) = rp (x, x). Thus the cIassifieation of hermitian inner produet spaees over F is reduced to the cIassifieation of quadratie inner produet spaces over Fo. 116 Appendix 2. Hermitian Forms
Proof by inductioll 011 the rank. First note that the field F eontains an element ::Lu with:xo +:x~ =F O. In the eharaeteristie 2 ease, :x u ean be any element in the eomplement of F;); while in the eharaeteristie =F 2 ease wc ean take ::L o = 1. Next note that for any hermitian inner produet spaee X over F, thc assoeiated bi linear form (xly)= is an inner produet over F;). In fact, given x =F 0 we ean eertainly ehoose x' so that (xlx')=:xu+:X~ =FO. Suppose now that the hermitian inner produet spaees X and Yare isomorphie as Fa-quadratie spaees. Sinee the assoeiated bi linear form (xix') is not identieally zero, we ean eertainly ehoose a veetor XE X and a eorresponding veetor yE Y with Q (x) = Q (y)=F O. Sinee Corollary. With F -:::J f~ as ab ave, lhere is all exacl sequence 0-> W(F,J)-> WQ(Fo)-> WQ(F) of W(Fu)-modules. Praa}: Given a non-zero element of W(F. J), we ean clearly ehoose a representative hermitian inner produet spaee X which is anisotropie: It follows that the underlying quadratie spaee is also anisotropie, and henee represents a non-zero element of WQ (Fu)' Thus the natural homomorphism W(F, J) -> WQ(Fu) is injeetive. Next let us look at the eomposition W(F, J) -> H"Q (Fo) -> WQ (F). Choose a basis {1,:x} for F over Fo. First eonsider a hermitian spaee X ofdimension lover F. Then X has a basis veetor 1.'1 with (lt may be eonvenient to ehoose :x so that :x +:xJ = 1 in the eharaeteristie 2 ease, or :x + 'Y. J = 0 in the eharaeteristie =\= 2 ease; but this is not neeessary.) Now let us pass to the indueed quadratie space F &; X of dimension F() 2 over F. Consider the non-zero veetor :X0CI -10c2 in this indueed spaee. Evidently 2 Q('Y.0CI -10(2 )=:x Q(cI)+Q(eZ)-'Y.(elieZ) =(:x 2 +:x:xl -:x(:x+:xl )) Q(cI)=O. Therefore this indueed quadnnie spaee is split. Sinee any hermitian inner produet spaee over F is c1early an orthogonal sum of I-dimensional spaees, this proves that the eomposition WCf~J)---->WQlf;J)---->WQ(F) is zero. COl1\ersely let Y be a quadratie spaee over Fo whieh maps to zero in WQ (F). After eliminating any split orthogonal summand, we may assume that Y is anisotropie. Sinec F 0 Y is split, it eertainly eontains a F" veetor y =\= 0 with Q(y) = O. Setting y=:X0Yl -1@>-'2 Substitllting :x 2 =:x (:x + :xl) -:x:xl and reealling that and :x are linearly independent over F;l' it follows that (YI!Y2) = (:x + :xl) Q(YI) Q(Y2)=:x:xl Q(YI), where QCI'I) =\= 0 sinee Y is anisotropie, This proves that the subspaee of Y spanncd by YI and .h is isomorphie to the underlying quadratie spaee of a hcrmitian spaee over F spanned by a veetor .11 \\ith !P(Yl' YI) = Q(y'I)' where :XYI eorresponds to the veetor Jz. Now express Yas an orthogonal sum (F;JYIt&FoY2)q)(F;JJIfBF;JY2)-L· The seeond summand has smaller rank, is also anisotropie, and also represents an element of the kerne!. A straightforward induetion now eompletes the proof. 0 Here are some examples worked out in more detail. (Compare [Milnor, 1969].) In cach case \Ve aSSllIIJe thm the inwlurion J is /lot the identity. EXllmple I. If Fis a finite field, then a hermitian inner produet spaee splits if and only if it has even rank. Thc rank is a eomplete invariant; and lV(F. J) ~ Z/2. 118 Appendix 2. Hermitian Forms Note that the deseription is exaetly the same whether the eharaeter• istie is 2 or *2. The proof will be left to the reader. Example 2. If F is a loeal field, or a funetion field in one variable over a finite field, then the rank and determinant of a hermitian inner produet spaee form a eomplete system of invariants. The kernel of the rank homomorphism W(F, J) -> Z/2 is an ideal, additively isomorphie to F;/norm F", and with square equal to 0. Again the eharaeteristie 2 ease is not distinguished in any way. The proof ean be sketehed as folIows. It suffiees to note that any spaee X of rank ~2 over F has rank ~4 over Fo, henee the quadratie equation Q(e1)= 1 has a solution. (Using the Hasse-Minkowski theorem in the global ease; eompare Chapter II, § 3. For the eharaeteristie 2 ease, see [Arf, pp. 164-167].) Henee X is isomorphie to an orthogonal sum (F e1)EIJ(F e1).l. Continuing induetively, we find an orthogonal basis e1, ... ,en with cp(e p e;}=Q(e;}=1 for i (J.2 _ ß2 ;2 _ y2 / + [)2 i2/. If ~e *0 for ~ *0, or in other words if the associated inner produet spaee Appendix 2. Hermitian Forms 119 over R o is anisotropie, then c1early R is a division algebra ( = skew field). In this ease, Jaeobson's argument applies just as before. There is a eanonieal embedding 0--> W(R,J)--> WQ(Ro}, and two hermitian inner produet spaees are isomorphie if and only if their underlying quadratie spaees are isomorphie over Ro. Jaeobson remarks also that the "determinant" of a hermitian spaee ean still be defined, in this non-abelian eontext, as an element of R~/norm (W). The definition is based on work of E. H. Moore. (Compare [Dyson].) Appendix 3. The Hasse-l\linkowski Theorem The Hasse-Minkowski theorem is one of the most beautiful results in algebraic number theory. The proof which follows assurnes some knowledge of Class Field Theory, as described in [Lang] or [Cassels• Fröhlich]. In the ca se of the rational field, it is possible to give a more elementary proof. (See [Serre] or [Borevich-Shafarevich].) For a complete and self-contained proof in the general case. see [O'Meara]. First some definitions. Let X be an inner product space over a field F. Then X is said to represel1t a field element 'J. if there exists a non-zero vector XE X with x· X =7.. Henceforth we assurne that F has characteristic =1= 2. Lemma 1. IJ u space X over F represel1ts 0, thell it represents every element oJ F. For if X represents ° then X admits a hyperbolic plane as direct summand (Chapter I, § 6), and a hyperbolic plane clearly represents all field elements. 0 Corollary. Aspace X represents the elemel1t 7. =1= 0 if al1d only if the orthogonal SUI11 X (j) <-7.) represents 0. (In other words an inhomogeneous equation in 11 nuiables can be expressed as a homogeneous equation in n + 1 variables.) The proof is immediate. 0 Now suppose that F is aglobai field. That is, F is either finite over the rational numbers, or finitely generated of transcendence degree I over a finite field. We continue to assurne that F has characteristic =1= 2. For every (non-trivial) valuation v of F, let F, denote the completion, and let X" denote the induced inner product space F'.:®X over F,.. Let 7. be some fixed element of F. Hasse-Minkowski theorem. The inner pl'oduct space X represellts 7. iJ ami ollly iJ X" represents CI. JOI' every (non-trivial) valuatiol1 v oJ F. Both archimedian and non-archimedian valuations v must be in• cluded. Appendix 3, The Hasse-Minkowski Theorem 121 In order to prove this theorem, it will be convenient to break it up into two parts. according as the field element I:J. is zero or non-zero: Assertion An' Let I:J.EF", Aspace X of rallk Il over F represents I:J. U' ([nd ollly i{ Xl represents rx jor every v. Using the corollary to Lemma I above, this is completely equivalent to the following statement. Assertion A~. Aspace Y o( rank /1 + lover F represents °if and 0/1/.1' U the completiolJ Yt, represellts °jiJr every valuation v. Note the shift from dimension n to 11 + I in Assertion A;,. In order to prove these two statements, we will pass back and forth between the two forms, first proving A 2 , and then showing that for n~4. The proof of AI will be given last, since it is completely irrele• vant to the rest of the argument. Proof of A z. Suppose that X ~ where rx*O. Alternatively, setting u= -UZ/ul and /3 = I:J., 11 1 , this can be written as (1) Let K denote the extension field F("~/u) = FeIl - u)uI ). Then the Eq. (I) possesses a solution ~, YJEF if and only if ß belongs to the image of the norm homomorphism normK / p : K" ~ F". If K*F this iscIear,since norm(~+YJ"0~)=e-1J2u.lf K=F it is cIear since 1I2 E-1II F"z, so the inner product space X splits and the Eq.(I) always has a solution. Now recall the foIIowing: Hasse norm theorem. Let K be a cyclic Ga/ais extension oI the global field F. 77len all element rx 0/ F" belongs to the image 0/ the homomorphism norm = normK.p : K" ~ F" if and only if rx belongs to the image 0/ jor aerr willatioll W of K. This is proved in [Lang, p. 195] or [Cassels-Fröhlich, p. 185]. 122 Appendix 3. The Hasse-Minkowski Theorem If the Eq. (1) has a solution in F,. for every v, then ß is a norm from K~v for every w, henee ß is a norm from K by the Hasse norm theorem. This eompletes the proof of Assertion A 2 . For the proof of A 3 , the following will be needed. Lemma 2. Let X be an inner product space of rank 3 and determinant d over a field F of characteristic =1= 2. To any field element et =1=0 we associate the extension field K = F(V - exd). TIlen X representsJ. if and only if the induced inner product space K ® X over K represents O. F Proo! If X represents ex, then X~(ex>$(ß>$(Y> for some elements ß, y, where d E ex ßy Fo 2 . Over the extension field K = F(V - exd) = F(V - ßy), we see that y is equal to -ß multiplied bya square. Therefore the spaee K®(ß>$(Y» splits, henee K ® X represents O. Conversely suppose that K ® X represents O. We may suppose that K is a proper extension of F, sinee if K = F then X itself represents 0, henee X represents et. Thus there exist two veetors x, YEX, not both zero, so that (x+V -etd y). (x+V -exd y)=O. In other words x· x-exd y. y=O, x· y=O. We may assurne that the inner produet x· x = etd y . y is not 0, sinee otherwise X would represent O. Henee the two orthogonal veetors x and y form part of an orthogonal base x, y, z, with X ~ (x· x>$ (y. Y>$ (z· z>, and d E (x . x)(y . y) (z . z) FO 2. Substituting the equation x· x = etd y . y, we see that z· z is equal to ex multiplied by an element of Fo 2 . This proves Lemma 2. 0 Proof that A~ implies A 3 • Fixing the spaee X of rank 3, and the field element et =1= 0, let K = F(V- exd) as in the lemma. F or eaeh valuation w of K we c1early have where v = w[F. Thus if X" represents r:J. for every L', then (K ® Xt re• presents 0 for every w. Using the Assertion A~, whieh has already been proved, it follows that K ® X represents 0, henee X represents CI.. Proof that A 3 implies A~ for n:S 4. Let X be a spaee of rank n + 1 ~ 5. Then X is isomorphie to a sum Y$Z where Z~(Ul>$(U2>$(u3> has rank 3, and Y~(u;>$"'$(U;I_2> has rank ~2. Appendix 3. The Hasse-Minkowski Theorem 123 Let T = T(u l , uz , Ll 3 ) denote the finite set consisting of all (equivalence dasses of) valuations v such that either (1) v is archimedian or dyadic, or (2) IUII,.=I= 1 or IUzl,=I= 1 or lu 3 1, =1= I. For v~ T we see as in Chapter Ir, § 3.4 that the completed space Z, necessarily represents 0. Suppose now that X, represents 0. Then we can certainly choose vectors y,.EY, and Z,EZ" not both zero, so that Y"·Yt+2,·z,.=0. In fact these vectors can be chosen so that Yt ' y,.= -z,.· z,,=I=O. For if our first choice of y, and z,. yields y,.' y, = z,' Z" = 0, then either Y represents 0, in which case we can choose an arbitrary z:. with z:· z: =1= ° and apply Lemma 1, or Z represents ° in which ca se any y;. y;. =1= 0 will do. We will also make use of the following. Weak approximation theorem. Given finitely many inequivalent valuations VI' ... , v, on a field F, the image of the diagonal embedding F ~ F;, x ... X F:, is e:;erywhere den se. This is proved for example in Lang, Algebra, Addison-Wesley 1965, p.285. Now consider the (n - 2)-dimensional vector space Y over F, and the set T = {VI' ... , v,}, Applying this approximation theorem to each of the n - 2 coordinates, we see that the image of the diagonal embedding Y~ Y,'J x ... x Y" is dense. In particular, we can choose an element YE Y which is so dose to y, for each VE T that the ratio (y . y)(y,,' yJ is a square in F;.". Let us apply Assertion A 3 to the space Z. For each VE T the com• pletion Z, represents - y,,' y,. and therefore represents - y . y. But for each v~ T the completion represents 0, and therefore represents - y . y. Applying A 3' it follows that Z itself represents - y . y, and therefore X = YEE> Z represents 0. This completes the proof of An and A~ for n ~ 2. To condude the proof of the Hasse-Minkowski theorem. we must prove Assertion Al' Given X~, and given:X=l=O in F, we must solve the equation Setting :x/tl = ß, this can be written as e = ß. Thus we must prove the following. 124 Appendix 3. The Hasse-Minkowski Theorem Square theorem. Il the jield elemellt ßE r is a square in fhe COH!• pletiol1 F;. Jor every v, then ß is a square in F. This theorem follows easily from the basic inequalities of global cIass field theory. Recall that the idele grollp A~ is the group of units in the ring A Fe n F". consisting of all elements (aJ in the cartesian product wh ich satisfy the condition la,.I,.~ 1 for almost all v. (In forming this cartesian prodllct, one of course chooses just one valuation v in each non-trivial equivalence cIass of valuations.) The quotient A;./F" is called the idele dass group Cl" For any finite extension K=:>F, the local norm homomorphisms K: ...... F.:1F combine to yield the global norm ho• morphisms A~ ...... A;., and CK ...... CF' If K is cycIic of degree m over F, then the inequalities of cIass field theory state that the index oJ the sub• grollp normK/F C K C CF is equal to m. See [Lang, p. 192] or [Cassels• Fröhlich, p. 179]. ProoJ oJ the square theorem. Gi yen ßEr, let K = Fc{lh If ß is a square in F;. for every v, then K w = FW1F for every valllation W of K, so the norm homomorphism A~ ...... A~ is surjective. Therefore the norm homo• morphism CK ...... CF is surjective, and the degree /11 must be 1. Thus Vß E F, which completes the proof of the square theorem and the Hasse• Minkowski theorem. [J Now consider a more general situation. Let X and Y be two inner prodllCt spaces over a field F with rank (X) ~ rank (Y). Definition. The space X is said to represent Y if X ~ Y $ Z for some Z. If Y has rank 1, say Y ~ X,.~ r..$Z(v) ~ r.:$ Z:.~ Thus Z;. represents LI for every v. By the Hasse-Minkowski theorem, it follows that Z' represents LI, say Z'~~Z". Together with (2), this completes thc proof. 0 Corollary 2. Two spac('s X (Illd Y over F are isomorphie i( and only i( X, is isomorphie to Y, jör ('very v. In partieular X splits if and ollly ij X" splils for erery v. Proo/. This is just the special case rank (X) = rank (}') of Corol• lary 1, using Chapter I. S 6.3. n Here is still another formulation of the Hasse-Minkowski theorem. Corollary 3. Consider a quadratie equation I tXij (i (j + I ßk (k + Y= 0, ill n variables, \Vith eoeff'icients in the global field F. I!, this equation has a solution in F;. Jor every v, then it has a solution in F. Note that the corresponding statement for equations of higher degree, or for systems of quadratic equations, would be false. Here is a trivial examplc. The 6-th degree equation has a solution in Q" for every v, but has no rational solution. Similarly, consider the simultaneous quadratic equations (2+1]=0, (1]-0(1]-(+16)=0, (2 = 172. For any solution ((, '1, 0 we must have (= ± 17 hence I] = 1, 17, -17 or - 33, and (2 + I] = 0. Again there is a solution in Q" for every v, but no solution in Q. Proo!, oJ Corol/ary 3. Write the given equation as (3) IX (x, x)+ ß(x)+y=ü, where tX is asymmetrie bilinear form on the vector space X = F", and where ß is an element of the dual vector space Hom(X, F). Let N be the null space of the linear mapping Xf->Ll(X, ) from X to Hom(X,F). Counting dimensions, we see that the sequence 0--> N --> X --> Hom(X, F) --> Hom(N, F) --> ° is exact. 126 Appendix 3. The Hasse-Minkowski Theorem If the element ßE Horn (X, F) restriets to a non-zero element of Hom(N, F), then it is easy to choose xEN so as to satisfy the required equation (3') O+ß(X)+y=O. Suppose then that ß maps to zero in Hom(N, F). Then ß lifts to an element of X, say for every XE X. The substitution x = y - X o now reduces Eq. (3) to the form (4) ~ (y, y) = y', with y'=ß(xo)-~(xo,xo)-Y. If N=O, so that X is an inner product space, we can apply the Hasse-Minkowski theorem to Eq. (4) and obtain a solution. If N =1=0, we must simply choose a complementary direct summand X=N$X'. Then ~ restricted to X' is an inner product, so the Hasse-Minkowski theorem applies, and Eq. (4) has a solution YE X' c X. This completes the proof of Corollary 3. 0 Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity Let L be a free Z-module of rank n, provided with a Z-valued sym• metrie bilinear form X· Y with non-zero determinant. We denote the signature of this form by a, and the absolute value of the determinant by d. An expression for exp(2n i (/8) as a finite exponential sum was given by H. Braun in 1940. (Explieitly, she showed that (2 at/2 yd exp(2n i (/8) = L exp(2n ix· xla) xEL/aL where a = 8 d3 . It folio ws that a mod 8 is determined by the an numbers X . x modulo a.) We will deseribe a c10sely related formula whieh has reeently been obtained by J. Milgram. (Compare the discussion In Cl:apter II, § 5.) As in Chapter II, we will say that L is of type II if the eongruenee (1) x·x=O mod2 is satisfied for every XE L. Let L# denote the dual lattice, eonsisting of all uEQ®L satisfying the eondition U· LcZ. Then the quotient L* I L is a finite abelian group of order d. If L is of type II, then setting c,o(U)=1U'U moduloZ, we obtain a weil defined quadratic funetion c,o: L# IL~ Q/Z. Theorem (Milgram). 1f L is of type II, then the Gauss sum L exp(2nic,o(u)) UEL# /L is defined and is equal to yd exp(2n i a/8). The original proof of this formula was a rather delieate argument involving the Poisson summation formula. The following proof, suggested by Knebusch. is quite a bit easier. Consider lattices L of type n in a fixed rational inner produet space. We will denote the d-fold sum Lexp(2nic,o(u))=Iexp(niu·u) briefly by the symbol G(L). I.#/L L"/L 128 Appendix 4. Gauss Sums. the Signature mod 8, and Quadratic Reciprocity Lemma 1. If LI cL is a sub-lattice a{ index k, thell G (LI) = k G (L). Praa! Evidently LI cL C L# c L~ where the index of each lattice in the next is equal to k, or d(L), or k respectively, Let x 1 "'" Xkd\L) be a complete set of coset representatives for L~ modulo L. Then the Gauss sum G (L 1 ) can be written as G(L1)=I I exp(2nicp(xj +ll)) . .\) IIE [ LI If we substitute this becomes G(L1)=Iexp(2nicp(xj )) I exp(2ni(xj '1l)), Xj UEL/L 1 But for each fixed xj the k-fold sum (2) I exp(2rci(xj 'u)) UEL,Lt can be evaluated as folIows. If xj happens to belong to L#, this sum is evidently equal to 1+", + 1 = k. If xjr/: L# then the correspondence llf---> exp(2rc i(xj Oll)) defines a non-trivial homomorphism from L/L1 to ce, so a standard argument [Lang, p, 82] shows that the sum (2) is zero, Therefore G(LI ) is equal to I exp(2rci cp(x k ))k=kG(L) as asserted, 0 2 Evidently d(L1 )=k d(L); so it follows from Lemma I that G (L)/Vd(L) = G(L1 )/Vd(L I), In fact the complex number G(L)/V d(L) is completely independent of the lattice L, and depends only on the ambient rational inner product space. This is clear, since any two lattices Land I.:. spanning the same rational space must contain a common sub-lattice Ln I.:. which has finite index in each of them, To evaluate this invariant G(L)/V d(L) of the inner product space Q® L, we recall that every rational inner product space is isomorphic to an orthogonal sum of I-dimensional spaces, and hence contains a lattice of type II which splits as an orthogonal sum of I-dimensional lattices, Note that the invariant G(L)/Vd(L) is multiplicative with respect to orthogonal sums, For the identity Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity 129 is easily verified, and the identity is familiar. Thus to compute this invariant G(L)/V d(L) for any rational inner product space, it suffices to compute it for a I-dimensional inner product space. The following elementary observation will be needed for the com• putation in the I-dimensional case. A Lemma 2. F or any constant c > 0, the integral Jexp (c 11: i S2) ds tends to a weil defilled finite limit as A ----> 'lJ. 0 To prove that the imaginary part of this integral converges, sub• stitute u = c S2 and integrate between successive integer values of u, noting that the terms of the resulting se ries alternate in sign, with absolute values tending monotonely to zero. Convergence of the real part is proved similarly, using half-integer values. 0 Consider now aI-dimensional lattice of type H, say L~(2m>. Suppose, to fix our ideas, that m > O. Evidently L* /L is cyclic of order 2m, and G(L) is equal to the 2m-fold sum 2m-l L exp(n i k2 /2m). k=O To evaluate this sum, following Dirichlet and Landau, we introduce an associated periodic function I: R ~ C of period 1, where 2m-l I(t)= L exp(n i(k+t)2/2m) k=O for O~ t~ 1. Thus 1(0)= 1(1) is equal to the Gauss sum G(L). (Compare [Lang, p. 88].) Since 1 is continuous and piecewise smooth, its Fourier series expansion converges to 1 everywhere. (See, for example, Titchmarsh, Theory of Functions; or Courant and Hilbert, Volume 1.) We will write this Fourier series in the form 00 I(t)= La n exp(-2nillt), -00 where 1 an = JI(t) exp(2nint) clt. o GO Note in particular that the Gauss sum G(L)=/(O) is equal to L an' -00 130 Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity To evaluate the coefficient an' first substitute the definition of f(t), obtaining 2m-li ( (k+t)2 )) an = I S exp 2n i +nt dt. k=O 0 4m Next complete the square, so as to obtain the congruence (k + t)2 4m +nt=(k+t+2mn)2/4m (modZ), and then substitute s = k + t + 2 m n, This yields 2m-1 k+I+2mn an = I S exp(2nis2/4m)ds k=O k+2mn 2m(n+l) S exp(nis2 /2m)ds, 2mn Now sum over n, so as to obtain the formula 00 2 G(L)=Ian = S exp(nis /2m)ds, -00 where the improper integral is weil defined by Lemma 2, In fact, sub• stituting u=sIV2m, it follows that G(L)=G«2m») is equal to 00 ~ S exp(n i u2 ) du, - 00 Thus the ratio 00 G«2m»)/V2m= S exp(niu2 )du - a; is independent of m, To evaluate this integral, we simply evaluate the Gauss sum for the case m= 1, obtaining the identity G«2m»)/~=(1 +i)/V2=exp(2n i/8), Similarly G« -2m»)/~ is equal to the complex conjugate exp(-2ni/8). Thus we have shown that the invariant G(LllVd(L) is equal to exp (2 ni 0/8) for every I-dimensional lattice L. The corre• sponding formula for an orthogonal sum of I-dimensionallattices, and hence for an arbitrary lattice, now follows immediately. This completes the proof of Milgram's theorem. 0 The formula of Braun can be recovered from that of Milgram as folIows. Let L be any lattice in a rational inner product space, subject only to the hypo thesis that X· YEZ for x, YEL. As before we set d= Ideterminantl >0, and n=rk(L). Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity 131 Corollary. If q is a multiple of 2d, then L exp(n ix . x/q) = qnl2 Vd exp(2 ni 0"/8). xELlqL Evidently the formula quoted at the beginning of this Appendix is an immediate consequence, taking q = a/2. Proof of the corollary. Consider the new inner product q x . y on the lattice L#. Note that L#, with this new inner product, is of type II, and has duallattice equal to q-l L. Applying Milgram's theorem we obtain L exp(niqu.u)=Vqn/dexp(2niO"/8). UEq-t LIL# Now substitute U=q-l x and multiply both sides of this equation by d. The left hand side becomes d L exp(n i x· x/q) = L exp(n ix· x/q)' xELlqL# xELlqL since qL# contains qL as a subgroup of index d. This completes the proof. 0 Knebusch points out that Milgram's formula is dosely related to a version of the quadratic reciprocity law due to [Weil, 1964]. Given Las above, let (L# /L)p denote the p-primary component of the finite abelian group L# /L. Then L# /L decomposes as the orthogonal sum of the various (L# /L)p' hence the Gauss sum G(L) can be expressed corre• spondingly as a product TI G((L# /L)p), where all but finitely many of the factors are equal to 1. Similarly the order d of L# IL splits as the product of its p-primary components dp • Lemma 3. nIe ratio Yp(L)=G((L#/L)p)/VJ;, depends only Oll the p-adic completioll Qp® L of the inner product space Q® L. 11!e correspondence Qp®LI--> rp(L) gives rise to a homomorphism from the fillite additive group W(Qp) to the multiplicative group of roots of unity in C. Briefly we say that Yp is a character of the Witt group W(Qp)' The proof of the first statement is completely analogous to the proof of Lemma 1. One simply uses the p-adic integers Zp and the p-adic field Qp in place of Z and Q. To prove the second statement, suppose that the completion Qp®Lis a split inner product space. Then this co m- pletion has inner product matrix(~ ~) with respect to a suitable basis, and this basis spans a Zp-lattice which is self.. dual. But the existence of such a self-dual lattice implies that Yp(L) = 1. Since the function Yp is dearly multiplicative with respect to orthogonal sums, this proves the lemma. 0 132 Appendix 4. Gauss Sums. the Signature mod 8. and Quadratic Reciprocity Let us define YCfJ (L) to be the root of unity exp( - 2 ni 0,/8). Evidently this depends only on the real completion R® L of the inner product space Q®L. Weil reciprocity theorem. For any lattice L, the pradllct TI (p(Q®L) is eqllal t(l 1. p;;; 00 Praa! This follows immediately from Milgram's theorem. 0 Let us see what this reciprocity formula means in the rank 1 case. Suppose then that the lattice L is spanned by a single vector 11 , Suppose also, to fix oUf ideas, that 11 .11 =4m with m odd. We will write brielly L=(4m>. Lemma 4. The character (2 (4 m> is eqllal to exp (2 n i m/8). In fact L# IL is cycIic of order 14ml, generated by Id4m. Hence the 2-primary component (L# ILlz is cycIic of order 4, generated by Id4, with cp (/1/4) == m/8 (mod Z). It follows easily that 4 (2 (4m> = L exp(2 ni / m/8)1V4 j=1 = exp(2 ni m/8). 0 Next suppose that m is an odd prime p. Lemma 5. The character (p (4 p >is equal to exp (2 n i(1 - p)/8). Proof This follows from Lemma 4, by solving the reciprocity equation More gene rally suppose that m = pu, where II is relatively prime to p. Lemma 6. The character t p(4 pu> is equal to ( ~ ) ::p (4 p>. Here the Legendre symbol (ulp) is defined to be either + 1 or -1 according as u is or is not a quadratic residue modulo p. Praa! Proceeding as above, the p-primary component (L# IL)p is spanned by a vector II/p with cP (/I/P) == 2 u/p(mod Z). Hence p (p(4Pll>= Lexp(2n i(2u/lp)). j=1 If (u/p) = + 1, then evidently the expression u/ varies over all quadratic residues modulo p, taking each non-zero value twice, and taking the Appendix 4. Gauss Sums, the Signature mod 8. and Quadratic Reciprocity 133 value zero rnodulo p just onee. On the other hand if (u/p) = -1 then u/ takes eaeh non-residue value twiee, again taking the zero value onee. Sinee the surn of exp(2n i(2k/p)) over all residue classes k rnodulo p is zero, the eonclusion follows easily. (Compare [Lang, p. 85].) 0 Now let p and q be distinet odd primes. Applying the reciproeity forrnula Y2(L) 'p(L) yq(L) 1'",,(L) to the lattiee L=(4pq) we obtain the identity (:) (:) exp(2ni(p-1)(q-l)/8)=1. This is just the classical quadratie reeiproeity law. Concluding re mark. There is an analogous Weil reciproeity forrnula over an arbitrary nurnber field, whieh ean be derived frorn the rational reciproeity forrnula. (Compare [Seharlau, 1972] and [Knebuseh• Seharlau, 1971].) It takes the form nl'v(X)=l, v where X is an inner produet spaee over the nurnber field F, and v ranges over all valuations of F. Here I'v is defined as follows. Case 1. If v is a eornplex arehirnedian valuation, then I L' (X) = 1. Case 2. If v is areal arehirnedian valuation, then I'v(X)=exp( -2n i O"v(X)/8) where O"v(X) is the assoeiated signature. Case 3. If v is the p-adie valuation, where p is a prime ideal in the ring of integers D, then y.(X) is defined as the ratio Here L ean be any D-Iattiee in X satisfying tl . I E D for I E L, and L# is the duallattiee with respeet to the Q-valued inner produet x, Yf--+ traeeF/Q x . y. The Gauss surn G((L#/L)Q) is defined as the surn ofexp(nitraee(u.u)) over all u in (L# / L)p' . Now the reeiproeity law n I'v(X) = 1 follows easily frorn Milgrarn's v forrnula, applied to the inner produet traee(x· y) on L. 134 Appendix 4. Gauss Sums, the Signature mod 8, and Quadratic Reciprocity One special case of this reciprocity law is of particular interest. Suppose that representing an element in the ideal [2 (F) in the Witt ring. Then it is easily verified that Yv(X)= ± 1. If r::*C, then both Weil and Scharlau show that Yv(X)= -1 for suitably chosen "J. and ß. The correspondence "J., ß I--> }' v (( ("J.) - (1») (ß) - (1»)) = ± 1 is in fact the "Hilbert symbol" associated with the valuation v. The equation TIY,((("J.)-(1»)((ß)-(l»))=1 is Hilbert's form of thc C' quadratic reciprocity law. (Compare [O'Meara], as weil as Chapter III, § 5.4-5.9, and Chapter IV, § 4.4.) Appendix 5. The Leech Lattice, and Other Lattices in Dimension 24 We will eonstruet a self-dual uni modular lattiee LcR24 with the property that x . x ~4 for every x *0 in L. (Compare Chapter 11, §§ 6, 7.) The eonstruetion begins with the following eombinatorial statement. Let F~4 denote the veetor space over F2 consisting of all 24-tuples of integers modulo 2. Lemma. There exists a 12-dimensional subspace Sc Fr with the following property. F or every non-zero vector s = (SI' ... ,S24) in S, the number of components Si which are equal to one is at least 8, and is divisible by 4. Furthermore S contains the vector (1, ... ,1) consisting of 24 ones. Proof Following Leeeh, we will display S as the row spaee of an explieit 12 x 24 matrix. Let Adenote the symmetrie 11 x 11 matrix over F2 whose first row is 01101000 and whose remaining rows are obtained by permuting these entries eyclieally to the left. Thus eaeh row of A eontains 6 on es. Patient in• speetion shows that (i) each pair of distinct rows of A has precisely 3 ones in common (i.e., in the same eolumn). Let B denote the symmetrie 12 x 12 matrix 0 1 1 ... 1 B= 1 1 A 1 whieh is formed from A by adjoining the first row 0 1 1 1 1 1 1 1 1 1 1 1 and a eorresponding first eolumn. Using (i) we easily verify the following. 136 Appendix 5. The Leech Latticc, and Other Lattices in Dimension 24 (ii) 771e matrix B satisfies the identity B Z = BB' = 1. H ence B is non• singular, and any two rows of B are orthogonal with re5pect to the inner product r . r' = L ri r;. The block matrix is now the required 12 x 24 matrix of rank 12. Evidently the sum of all of the rows of C is eq ual to the 24-tuple (1, 1, ... , 1). Note that (iii) The number of ones in any row of Cis either 8 or 12. Furthermore any two di5tinct rows of C are orthogonal. It will be convenient to use the notation 11511 for the numbcr of ones in a 24-tuple 5=(051 ,,,,, S2 .. )' As a corollary of(iii) we obtain the following statement. (iv) 1f 5 is a linear combination of the rOW5 of C, thell 11511 =0 (mod 4). This is proved by induction on the number of rows involved. If s' is obtained from s by adding a row 1', then evidently 11.'i'11 = 11511 + Ilrll-2n where n denotes the nUll1ber of ones which 5 and r have in common. But 5 and r are orthogonal by (iii), so n is even. Assuming inductively that 11511 is divisible by 4, it follows that 115'11 is divisible by 4 also. (v) 1f 5 is a non-zero linear combination of the rOW5 of C, then 11 sll ~ 8. Proof By (iv), it suffices to prove that IIsll ~ 5. Suppose that 5 is the sum of k distinct rows of C. The case k = 1 is covered by (iii). If k = 2, then it follows easily from (i) that 11511 = 8. If k = 3 and if 5 is the sum of the first row of C and two other rows, then again it follows from (i) that Ilsll = 8. If 5 is the sum of three rows of C not including the first row, then evidently the first thirteen entries of s include precisely 4 ones. If the remaining eleven entries were all zero, this would mean that the sum of the three corresponding rows of A was zero. Hence the sum of the remaining eight rows of A would also be zero; and the sum of the corre• sponding eight rows of B would be zero, contradicting (ii). Therefore 11511 ~5. Finally, if k~4, then the first twelve entries of s contain at least 4 ones, and the remaining entries contain at least 1 one by (ii), so again it follows that 11511 ~5. This proves (v), and completes the proof of the Lemma. 0 Remark 1. The matrix C was constructed in a rather ad hoc mann er. The following description of its row space S may seem a !ittle more motivated. Consider the field Fzo .. s with 211 elements. We claim that S Appendix 5. The Leech Lattice, and Other Lattices in Dimension 24 137 ean be identified with the eolleetion of all "relations" between the rd 23 rd roots of unity in F21J4H . Let (f) denote a 23 root of unity satisfying the irredueible equation 1 +w+ws + 0)6 +w7 +w9 +Wll =0 over F2 , and let ep denote the Frobenius automorphism ::xc--+ (X2. Then the list ep'I(W-'), epllJ(W- 1), ... , (pl(U)-I), I, epl(W), ep2(W), ... , ep"(W) rd eontains eaeh 23 root of unity in F21J48 just once. Now S is the colleetic:n of all 24-tu pIes (SI' ... , S 24-) of in tegers mod 2, wi th sum zero, satisfying the linear relation S2 epll (w-') + S 3 ep IIJ (w- I ) + ... + S'2 ep' (w- I ) + SI 3 + Sl4 ep' (w) +"·+S24ep"(W)=0. Details will be omitted. Remark 2. There is an interesting simple group assoeiated with the matrix C. The Mathieu group M 24 ean be defined as the group of all permutations of the columns of C whieh transform eaeh row into a linear eombination of the rows of C. This group aet:; S-fold transitively, and has order 24·23·22·21·20·48. Although it was deseribed by Mathieu in 1861, its existence was first firmly established by Witt in 1938 (Harn burg Abh. 12). We are now ready to eonstruct the Leeeh lattiee. Let LIJ denote the 24 lattice in R with orthogonal basis h l , ... ,b24 where bi·bi=t. Let L denote the sublattice of LIJ consisting of all linear combinations t, b, + ... + t 24 b24 with integer eoefficients such that either (vi) the incegers t l , .•• , t24 are all even, with t l + ... + t 24 =0 (mod 8), ami the 24-tuple 1(t" ... , t 24) rechlced moduio 2 belongs to the vector space S oI the lemma; 01' (vii) the integers t l , ... , t24 are all odd, with t l + ... + 124 =4 (mod 8), and the 24-tuple 1(1 + tl , ... , 1 + t24 ) reduced modulo 2 belongs to S. An easy argument shows that L is closed und er addition, forming a sublattice of index 236 in L o. Therefore det L = 436 det L o = 1. We will show that the norm k(t~ + ... + t~4) oI an element oI L is always an even integer. In other words t~+ .. ·+t~4=0 (mod 16). If the ti are even, then tf is congruent to 0 or 4 moduln 16 aeeording as ti is eongruent to 0 or 2 modulo 4. But it follows from the lemma that 138 Appendix 5. The Leech Lattice, and Other Lattices in Dimension 24 the number of ti congruent to 2 modulo 4 is divisible by 4, If the t i are odd, then t; is congruent to I or 9 modulo 16 according as t i is con• gruent to ± 1 or ± 3 modulo 8. Thus It;=cx! +9cx 3 +9cx s+cx 7 (mod 16), where 'Y.j denotes the number of t i which are congruent to j modulo 8. Note the congruences IX!+cx 3 +CX S +cx 7 =24=0 (mod8), cx! +3cx 3 +5cx 5 +7cx 7 =4 (mod 8), =0 (mod 4), where the last two follow from (vii) and the lemma. Adding the first two congruences and subtracting twice the third, we obtain 4cx 3 +4cx s =4 (mod 8). Thus IX3 + IX s is odd, and it follows that It; = 24 + 8 (cx) + cx s) = 0 (mod 16). For any x and y in L, it follows that x· y=! ((x + y). (x+ y)-x· x- y. Y)EZ. Thus the la!tiee L is self~dual. Note that no element XE L can satisfy X . x = 2. For if ti + ... + t~4 = 16 then the l i certainly cannot all be odd. But the only expressions for 16 as a sum of even squares are 16=42 = 22 + 22 +22 + 22 , and both possibilities are excluded by (vi) and the lemma. Therefore x . x;?; 4 for every x =l= O. Für further information about the Leech lattice, the reader is referred to [Conway]. Concluding remark. A complete classification of unimodular lattices of type II in R 24 has been given by [NiemeierJ. He shows that there are precisely 24 such lattices L up to isomorphism; and that a complete in• variant for L is provided by the finite subset R (L) consisting of all vectors xEL with norm X· x equal to 2. Evidently, for any xoER(L) the reflection y~ y-(xo ' y)xo in the hyperplane perpendicular to X o maps L to itself, and hence maps R (L) to itself. Therefore R (L) is a .. root system ", as described in [Bour• baki, v.34, pp. 142-197], in some euclidean space. Note that the angle between any two vectors in R(L) is either 0°, 60°, 90°,120°, or 180°. Using the classification theorem for root systems, we see that R (L) is a disjoint union of mutually perpendicular root systems, each of which Appendix 5. The Leech Lattice. and Other Lattices in Dimension 24 139 can be described by a "Dynkin diagram " of one of the following three types. In each case, each vertex of the Dynkin diagram represents a basis vector of norm x . x = 2 in an m-dimensional lattice, and two such basis vectors have inner product either - I or 0 according as they are joined by a line segment or not. The associated root system consists of all vectors of norm 2 in the lattice L spanned by these basis vectors. Type A", (m ~ I). In this case the Dynkin diagram consists of l1J vertices joined by m - I line segments as folIows. In terms of auxilliary orthonormal vectors e t , ." , e m + l' the i-th vertex in this diagram can be identified with the difference vector ei - e i + 1 . Thus the lattice L can be identified with the lattice consisting of all (m + I )-tuplcs of integers with sum zero. The determinant of L is equal to m+ 1. Type D", (m ~ 4). In this case the m vertices are connected as folIows. In terms of orthonormal vectors e1 , '" , 1:'"" the m vertices can be iden• tified with the vectors I:'i-ei+l and 1:''''_1 +1:'",. The lattice L can be identified with the lattice consisting of all m-tuples of integers with even sum. Its determinant is equal to 4. Type E", (/11 = 6, 7, 8). In these three exceptional cases the m vertices are connected as folIows; and the determinant of L is equal to 9 - m. '''1-·-- I • Compare the discussion in Chapter II, § 7. Niemeier gives an explicit list of the 24 distinct root systems which arise from unimodular lattices of type II in R 24. In general the fOot system R (L) spans a sub-lattice L which has finite index in L. The Leech lattice, with L = 0, is the only exception to this. In general the lattice L has determinant greater than 1, and hence is a proper sub-lattice of L. Again there is just one exception, namely the lattice L=L=lsffilsffils with root system R (L) equal to Es u Es u Es. Note that L may be (and usually is) decomposable, even when the uni modular lattice L is in• decomposable. Chronological Table Adrien-Marie Legendre 1752-1833 Carl Friedrich Gauss 1777-1855 lames Joseph Sylvester 1814-1897 Charles Hermite 1822-1901 Gotthold Eisenstein 1823-1852 Leopold Kronecker 1823-1891 Henry lohn Stephen Smith 1826-1883 Aleksandr Nikolaevic Korkin 1837-1908 Arnold Meycr 1844-1896 Egor I vanovic Zolotarev 1847-1878 Hermann Minkowski 1864-1909 Carl Ludwig Siegel 1896- Helmut Hasse 1898- Ernst Witt 1911- Albrecht Pfister 1934- References Arason, J. K., Pfister, A.: Beweis des Krullschen Durchschnittsatzes für den Wittring. Inven!. Math. 12, 173-176 (1971). Arf, c.: Untersuchungen über quadratische Formen in Körpern der Characteristic 2. J. Reine Angew. Math. 183, 148-167 (1941). (Compare ibid. 193, 121-125 (1954).) Artin, E.: Geometric algebra. Interscience 1957. Bass, H.: Lectures on topics in algebraic K-theory. Bombay: Tata Institute 1967. Birkhoff, G., MacLane, S.: A survey of modern algebra. MacMillan 1941. Blichfeldt, H. F.: The minimum values of quadratic forms in six, scven, and eight variables. Math. Z. 39, 1-15 (1935). Blij, F. van der: An invariant of quadratic forms mod 8. Indag. Math. 21, 291-293 (1959). Borevich, Z. I., Shafarevich, I. R.: Number theory. Academic Press 1966. Bourbaki, N.: Elements 24 (Algebre 9), Formes sesquilineaires et formes quadratiqucs. Hermann 1959. Bourbaki, N.: Elements 34, Groupes et algebrcs de Lie 4-6. Hermann 1968. Br~'un, H.: Geschlechter quadratischer Formen. J. Reine Angew. Math. 182,32-49 (1940). Cartan, H., Eilenberg, S.: Homological algebra. Princeton University Press 1956. Chevally, c.: The algebraic theory of Spinors. Columbia University Press 1954. Conway, J. H.: A group of order 8,315,553,613,086,720,000. Bull. Lond. Math. Soc. 1, 79-88 (1969). Conway, J.H.: A characterization of Leech's lattice. Invent. Math. 7, 137-142 (1969). Conway, J. H.: Groups, lattices, and quadratic forms, pp. 135-139 of "Computers In Algebra and Number Theory", SIAM-AMS Proceedings 4, AMS 1971. Dickson, L. E.: History ofthe Theory ofNumbers, 2 and 3, New York: G. E. Stechart & Co. 1934. Dyson, F.J.: Quaternion determinants. Helv. Phys. Acta 45, 289-302 (1972). Fröhlich, A.: Discriminants of algcbraic number fields. Math. Z. 74, 18-28 (1960). (See also: Ideals in an extension field ... , pp. 29-38.) Fröhlich, A.: Hermitian and quadratic forms over rings with involution. Quart. J. Math. Oxford 20, 297-317 (1969). Fröhlich, A.: On the K-theory of unimodular forms over rings of algebraic integers. Quart. J. Math. Oxford, to appear. Fröhlich, A., McEvett: Forms over rings with involution. J. Algebra 12, 79-104 (1969). Gel'fand, I. M., Mischchenko, A. S.: Quadratic forms over commutative group rings and the K-theory. Functional Analysis and its Applications 3, 277-281 (1969). Geyer, W.-D., Harder, G., Knebusch, M., Scharlau, W.: Ein Residuensatz für symmetrische Bilinearformen. Invent. Math. ll, 319-328 (1970). Hilbert, D., Cohn-Vossen, S.: Geometry and the imagination. Chelsea 1952. Hirzebruch, F.: Topological methods in algebraic geometry. Springer 1966. Jacobson, N.: A note on hermitian forms. Bull. Amer. Math. Soc. 46, 264-268 (1940). Knebusch, M.: Grothendieck und Wittringe von nichtausgearteten symmetrischen Bi- linearformen. Sitzungsber. Heidelb. Akad. Wiss. Math.-naturw. KI. 196970, 3. Abh. 142 References Knebusch, M.: Runde Formen über semilokalen Ringen. Math. Ann. 193,21-34 (1971). Knebusch. M.: Ein Satz über die Werte von quadratische Formen über Körpern. Invent. Math. 12,300-303 (1971). Knebusch. M., Rosenberg. A., Ware, R.: Structure of Witt rings. quotients of abelian group rings, and orderings of fields. Bull. Amer. Math. Soc. 77, 205-209 (1971). Knebusch, M., Rosenberg, A., Ware, R.: Structure of Witt rings and quotients of Abelian group rings. Amer. J. Math. 44,119-155 (1972). Knebusch, M., Rosenberg, A., Ware, R.: Grothendieck and Witt rings of hermitian forms over Dedekind rings. Pacific J., to appear. Knebusch, M., Rosenberg, A .. Ware, R.: Signatures on semilocal rings. Bull. Amer. Math. Soc. 78, 62-64 (1972). Knebusch, M., Scharlau, W.: Über das Verhalten der Witt-Gruppe bei Körpererweiterun• gen. Math. Ann. 193, 189-196 (1971). Knebusch, M., Schar lau, W.: Quadratische Formen und quadratische Reziprozitäts- gesetze über algebraischen Zahlkörpern. Math. Z. 121,346-368 (1971). Kneser, M.: Zur Theorie der Kristallgitter. Math. Ann. 127, 105-106 (1954). Kneser, M.: Klassenzahlen definiter quadratischer Formen, Arch. Math. 8, 241-250 (1957). Korkinc, A, Zolotareff, G.: Sur les formes quadratiques positives. Math. Ann. 11, 242-292 (1877). Landherr, W.: Äquivalenz Hcrmitescher Formen über einen beliebigen algebraischen Zahlkörper. Abh. Math. Sem. Univ. Hamburg 11. 245-248 (1935). Lang. S.: Algebraic number theory. Addison-Wesley 1970. Leech. J.: Notes on sphere packings. Canad. J. Matn. 19, 251-267 (1967). Leech.1.. Sloane, N.J. A: New sphere packings in dimensions 9-15. Bull. Amer. Math. Soc. 76. 1006-10 10 (1970). Lorenz. F.: Quadratische Formen über Körpern. Lecture Notes in Mathematics 130, Springer 1970. Lorenz. F., Leicht, J.: Die Primideale des Wittschen Ringes. Invent. Math. 10, 82-88 MacKenzie, R .. Scheuneman, J.: A number field without a relative integral basis. Amer. Math. Monthly 78, 882 (1971). MacLane. S.: Hamiltonian mechanics and geometry. Amer. Math. Monthly 77, 570-585 (1970). Mars. J. G. M.: The Siegel formula for orthogonal groups, Proceedings Symposia in Pure Math. 9, AM.S. 1966, pp. 133-142. Massey. W.: Algebraic topology. Harcourt, Brace and World, 1967. Meyer. A: Über die Auflösung der Gleichung ax2 +b y2 + c 2 +dL/ 2 +ev2 =0 in ganzen Zahlen. Vierteljahrs. Naturfors. Gesells. Zürich 29, 220-222 (1884). Milnor. J.: On simply connected 4-manifolds, Symposium Internacional Topologia Alge- braica. Mexico, 1958. Milnor, J.: On isometries of inner product spaces. Invent. Math. 8. 83-97 (1969). Milnor. J.: Algebraie K-theory and quadratic forms. Invent. Math. 9, 318-344 (1970). Milnor. J.: Symmetrie inner products in characteristic 2, in Prospects in Mathematics, Annals Study 70, Princeton University Press 1971. Milnor. J.: Introduction to algebraie K-theory. Annals Studies 72. Princeton University Press 1971. Niemeier, H.-V.: Definite quadratische Formen der Diskriminante 1 und Dimension 24. J. Number Theory (to appear). O·Meara. O. T.: Introduction to quadratic forms. Springer, 1963. Pfister. A.: Quadratische Formen in beliebigen Körpern. Invent. Math. 1, 116-132 (1966). Rogers. C. A.: The paeking of equal spheres. Proc. London Math. Soc. 8, 609-620 (1958). Rogers. C.A.: Packing and covering. Cambridge University Press 1964. Sah, Chih-Han: Symmetric bi linear forms and quadratic forms. J. Algebra 20, 144-160 (1972). References 143 Scharlau, W.: Quadratic forms, Queen's papers on pure and applied malh. 22, Queen's Univ., Kingston, Ontario, 1969. Schar lau, W.: Zur Pfisterschen Theorie der quadratischen Formen. Invcnt. Math. 6, 327-328 (1969). Scharlau, W.: Induction theorems and the structure of the Witt group, Invent. Math. ll, 37-44 (1970). Scharlau, W.: Hermitesche Formen über lokalen Körpern. Math. Ann. 186, 201-208 (1970). Scharlau, W.: Quadratic reciprocity laws. J. Number Thcory 4,78-97 (1972). Serrc, J.-P.: Cours d'arithmetique, Presses Univ. France, 1970. (English translation, to appear.) Siegel, C. L.: Gesammelte Abhandlungen I. Springer 1966, pp. 326-405. Spanier, E.: Algebraic topology. McGraw-Hil! 1966. Springer, T. A.: Quadratic forms over fields with a discrete valuation LIndag. Math. 17, 352-362 (1955). Steenrod, N.: The topology of fibre bundles. Princeton University Press 1951. Swan, R.: Vector bundles and projective modules. Trans. Amer. Math. Soc. 105,264-277 (1962). Swan, R.: Algebraic K-theory, Lecture Notes in Mathematics 76, Springer 1968. Waerden, B. L. van der: Die Reduktionstheorie der positiven quadratischen Formen, Acta Math. 96, 265-309 (1956). Wall, C. T. c.: Surgery on compact manifolds, Academic Press 1970. Watson, G. L.: Integral quadratic forms. Cambridge University Press 1960. Weil, A.: Sur certains groupes d'operateurs unitaires, Acta Math. llI. 143-211 (1964). Weil, A.: Sur la formule de Siegel dan, la theorie des groupes c1assiques. Acta Math. II 3, 1-87 (1965). W:tt, E.: Theorie der quadratischen Formen in beliebigen Körpern. J. Reine Angew. Math. 176,31-44 (1937). Zariski,O., Samuel, P.: Commutative Algebra I, Van Nostrand 1958. Index anisotropie space 56, 112 fundamental domain 15 Arason, J. K. 76 fundamental ideal I 66 Arf, Cahit 112 Artin-Schreier theorem 60 Gaussian integers 39, 95 asymptotic estimates 17, 31, 35, 50 Gauss sums 51, 127 genus of symmetrie bilinear form 43 Bachet de Meziriac 40 global field 120 bilinear form 1 bilinear form module or space 2 Hasse invariant H 79 Blichfeldt, H. F. 29, 35 Hasse-Witt invariant h 80 Blij, F. van der 24,25 Hasse-Minkowski theorem 20,89, 120 Braun, Hel 127, 130 Hasse norm theorem 121 Hermite, Charles 18 characteristic two 56, 82, 100, 112 hermitian form 114 convex set 16 Hilbert symbol 78, 134 convolution 55 Hlawka, E. 32,46 Conway, J. H. 28.46 hyperbolic plane 9.12-13. 101 cydic Will groups 23, 66, 69,87,99 Dedekind domains 7,91,93 I (fundamental ideal) 66 density of packing 34 In (genus of n E6 , E7 , Es (root systems) 28,31,139 involution 114 Eichier, Martin 27 Eisenstein, Gotthold 18,45 Jacobi, C. G.J. 23,45,61 Euler, Leonhard 39 Jacobson, Nathan 115.118 exterior power 12, 107 extreme lattice or matrix 29, 38 Klein bottle 101 Knebusch, Manfred 12. 14.85.95, 127, face centered cubic lattice 30,31,35 131 Fermat, Pierre de 39 Kneser, Martin 28 finite field Fq 21,81. 87,117 Korkine, Alexander ( = Korkin. Aleksandr) Fourier series 129 28,29,38 146 Index Lagrange, Joseph Louis 40 residue dass form 85 lattice 15, 91 Rogers, C. A. 31,35 Leech,John 28,35,135 root systems 31, 138 Legendre symbol 43,51, 132 Leicht, J. 65 Scharlau, Winfried 71,133 level S of a field 75 self-duallattice 46.91 local ring 6-8, 14, 68, 86 shoe box principle 21 Lorenz, Falko 65 Siegel, earl Ludwig 44-49 Lusztig, G. 106 signature a 23-25,62-64,69, 127 similarity transformation 29, 76 mass (of genus) 49 simple finite groups 28, 137 Mathieu group 137 skew bilinear form 2 Meyer, Arnold 20 split inner prod uct space 12 Milgram, James 25, 127 Smith, H.J.S. 28,41 minimal vector 27 Springer, T.A. 85 Minkowski, Hermann 16,20,31,41 square theorem 124 multiplicative inner product space 72 Steinberg, Robert 47,78,81 Steinitz, Ernst 7, 107 Niemeier, H.-V. 138 Stirling's formula 31 nilpotent elements, nilradical 68,69, 76, sums of squares 39.41,45,74 95 Sylvester, J.J. 23,61 norm (in various senses) 23,72, 115, 121 symbol 78 number field 61,64,81,94, 107, 118 symmetric bilinear form 2 ordering of field 59.67 symplectic basis 7. 106 orientation 94, 101,103 symplectic bilinear form 2,7, 105 orthogonal 2 tensor product 10,47,73,111 orthogonal sum, complement, basis 4-6 Thompson, John 46 packing of eudidean spacc 34 Thue, A. 35 p-adic integers Zp 21,25,42 torsion in Witt ring 68, 69, 72 v-adic numbers Qp 20,81,89 torus 16, 101 Pfaffian 7 totally real, imaginary 95 Pfister, A. 65, 72, 76 totally positive 61,95 positive definite 16,26,61 type I, type II 22-25 projective module 2 projective plane 101, 103 unimodular lattice 16 pythagorean field 71 valuation, valuation ring 84 Q, Qp 20 vector bundles 105-107 quadratic form 22, 110 volume 15,34,42 quadratic inner product space 111 Voronoi polyhedron 30 quadratic reciprocity 132-134 quaternion algebra 118 weak approximation theorem 123 Weil. Andre 131-134 Radon-Nikodym theorem 42 Witt, Ernst 8,28.84, 137 rank 3, 11 Witt ring, Witt dass 14, 112, 114 radical 68, 69, 95 Witt ring of Z 23. 90 rational numbers Q 20, 87 Witt rings of specialfields 81,87,88 real fields, real dosure 60, 71 reflection 7, 138 Z, Zp, Z", 15,42, 43 regular hexagonal lattice 30, 35 Zn 63,64 represent 120, 124 Zolotareff, G. ( = Zolotarev, E.l.) 28, 29, resid ue dass field 84 38 Special Notations (B), (u) (inner product space specified by matrix) 3,4 B' (transpose of matrix) 3 M.l (orthogonal complement) 5 R" (units of ring) 4 Tg (lattice) 27-29,31,35,44,47,103,139 W n (volume) 16,31 Ergebnisse der Mathematik und ihrer Grenzgebiete 1. Bachmann: Transfinite Zahlen. DM 48,-; US $15.30 2. Miranda: Partial Differential Equations of Elliptic Type. DM 58,-; US $18.40 4. Samue1: Methodes d'Algebre Abstraite en Geometrie Algebrique. DM 34, -; US $10.80 5. Dieudonne: La Geometrie des Groupes Classiques. DM 42,-; US $13.40 7. Ostmann: Additive Zahlentheorie. 1. 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