the zeta function of a birational severi-brauer scheme

BOL. SOC. BRAS, MAT,, VOL. I0 N ~ 2 (1979), 25-50 25

The zeta function of a birational Severi-Brauer scheme

Slltdrt 7-tlrner

~1. Introduction.

Let k be a field and V be an algebraic k-scheme of dimension n - 1. V is called a Severi-Brauer k-scheme if there exists a separable algebraic extension L/k such that VxkL and P , _ I ( L ) = Proj L IX 1 . . . . X , ] are isomorphic,as L-schemes ([11], p. 168). V is said to be split by L/k. V is called a trivial Severi-Brauer k-scheme if V and P ,_ l(k) are isomorphic as k-schemes.

Let .K/k be a finite Galois extension and let G = Gal (K/k). The isomorphism classes (as k-schemes) of Severi-Brauer k-schemes of dimension n - 1 which are split by K/k are in canonical one-one corres- pondence with the elements of the cohomology set HI(G, PGL(n, K)) ([1, 1], loc. cit.).The exact sequence 1 --* K* ~ GL(n, K)~ PGL(n, K)~ 1 induces a map A: HI(G, PGL(n, K))--.H2(G, K*). A is injective and Im A is described as follows: let 7 ~ H2( G, K*) and let D(7) denote the central division algebra over k defined by 7, let [D(7): k] = d 2, then 7 s Im A if and only if din ([10]). Assume now that 7 ~ I m A and let f l=A- l (7) , then the Severi-Brauer k-scheme V(fl) defined by fl is isomorphic as a k-scheme to the Grassmann variety of left ideals of rank n in the matrix algebra A = M,,,I(D(7)). Actually, A does not depend on K but only on n and the image of 7 in Br(k), the Brauer group of k. It was Chatelet ([6]) who first defined Severi-Brauer varieties and established their basic properties over fields of characteristic zero. Inpa/'ticular, he determined the function field, K(V(3)), of V([3). Amitsur ([2]) interpreted Chatelet's work from another point of view, obtained new results, and extended Chatelet's theorems to fields of arbitrary characteristic. Today, his determination of K(V([~)) is more accessible than Chatelet's. The defining relations for K(V(/~)) are simplest when A is a cyclic algebra ([1,7]) i.e. there exists

AMS Subject Classifications (1970). Primary: 12A80, 14DIO, 14GlO, 14G25. Secondary: 12A65. 12B35, 14G20.

This research was supported in part by the NSF (GP 7952X2 and GP 7952X3), BNDE, CNPq, FINEP, and OAS!

Recebido em fevereiro de 1979.

26 Stuart Turner

a cyclic splitting field k'/k for A with [k': k] = n. In this case, let z ~ k' such that k ' = k(z). Chatelet and Amitsur proved that there exists a 0 ~k* such that K(V(fl)) is k-isomorphic to the homogeneous quotient field of M = k [Xo . . . . , Xn]/I, where I denotes the ideal generated by

0(Xo .... , x o ) = I-I (Xo+o(z)Xl+o(z2)X2+...g(z~-1)x,-~)-OX~. g~Gal(k'/k)

Since the coefficient of each monomial of g(X o , X I . . . . . X,,) is a symmetric polynomial in {z, o'(z) . . . . , o"-l(z)}, it is clear that each coefficient is actually an element of k.

The element 0 arises in a natural way. To see this let d be the group of characters of G = Gal (k'/k). There is a canonical pairing

~: d x k* ---, H2(G, k'*) ~ Br(k)

([11], p. 211; [15], p. 181). Let Z be a generator for d and n~: k* --* H2(G, k'*) be the homomorphism defined by r/z(b)= r/0~, b). Ker ~/x consists of the elements of k* which are norms from k'*. By construction 7 ~ l m ~x" For 0 one can choose any element of k* such that r/(X, 0) = 7- As there may exist many cyclic splitting fields for A of degree n over k, the repre- sentation given by Chatelet and Amitsur of K(V(/~)) as the homogeneous quotient field of the grade k-algebra M depends on the choice of k'. It also depends on the choice of Z, on the choice of the primitive element z for k'/k, and on the choice of 0. For an arbitrary field k there is no natural way to make these choices. However, in case k is a locally compact field, distinct from • or C (a p-field in the terminology of [15]) or in case k is an algebraic number field it is possible to use the result of Chatelet and Amitsur to construct projective models for K(V(3)) which are defined over the ring of integers in k. The construction of these models is carried out in w and w

It will be convenient; particulary in ~ , to avoid mention of the algebras by using the following.

Definition. Let 7 ~ Br(k) and iet D(~) denote the central division algebra over k defined by 7. ~ will be called a cyclic Brauer class of degree n in case there exists a cyclic extension k'/k of degree n such that D(7)| is a full matrix algebra over k'. k' will be called a splitting field for ~ of degree n over k.

As it will be necessary to work in a "coordinate-free" context, we make the following.

The zeta function of a birational Severi-Brauer scheme 27

Definition. Let V(fl) be a Severi-Brauer k-scheme of dimension n - 1 such that A(fl) is a cyclic Brauer class of degree n. Let k' be a splitting field for ~ of degree n over k. Let G be the group of characters of G = = Gal(k'/k) and let X be a generator for G. Let z ~ k' such that k ' = k(z) and 0 ~ k* such that r/(Z, O) = A(fl) ~ H2(G, k'*). Let M = k[X o,. . . ,Xn]/I, where I is the ideal generated by

I-I (x0 + ~(z) x l + g(z ~) x2 + ... + g(z n- 1) x~_ 1) - 0 x~. geG

Let M ~ denote the quasi-coherent grade Ospeck-algebra defined by M. A k-scheme k-isomorphic to Proj (M ~) will be called a Chatelet model for V(fl). /

It is irdportant to observe that Proj (M "~) is birationally, and not biregularly, isomorphic to V(fl). Indeed, there are singular points on the "hyperplane at infinity".. Their existence is reflected in the results of w where the" specializations of models for Severi-Brauer schemes defined over discrete valuation rings are determined.

w begins with the study of a non-trivial Severi-Brauer scheme V(fl) defined over a p-field. A projective model for V(fl), defined over the ring of integers of the field, is constructed. Then trivial Severi-Brauer schemes defined over an arbitrary field with a discrete valuation are studied. Two types of models for these schemes are constructed. In w the specializations of the models given in w are determined and in case the discrete valuation ring has finite residue field the zeta function of the specialization of each model is computed. Severi-Brauer schemes defined over an algebraic number field k are investigated in ~ and w ~ contains a detailed analysis of cyclic Brauer classes of degree n. The arguments are based on the Grunwald-Wang theorem and on strong approximation. As it contains results in class field theory which seem to be new and may be of interest apart from their application to Severi-Brauer schemes, we have written this w so that it can be read independently of the rest of the paper. w contains the construction of a projective model of a Severi-Brauer k- scheme that is defined over the ring of integers of k. The basic properties of this model are established. In w the zeta function of this model is calculated and its functional equation is established.

Much of the work on models for Severi-Brauer schemes was begun at The Institute for Advanced Study during 1970-72 where the author profitted from A. Weil's advice and encouragement. Besides thanking Weil, the author also wishes to thank J. -L. Verdier for helpful discussions. He is also grateful for the hospitality extended to him by Brandeis Uni- versity, and the Universidad Nacional Aut6noma de M6xico.

28 �9 S tua r t T u r n e r

w Models for Severi-Brauer schemes over local fields.

Let k be a p-field with ring of integers r and residue field Fq. Let V(fl) be a non-trivial Severi-Brauer k-scheme of dimension n - 1. By local class field theory V([1) is split by k,/k, where k, denotes the unramif ied extension of k of degree n. k, contains a primitive q " - 1 root of uni ty

and k, = k(O. Let G = Gal(k./k) and G denote the group of characters G. There is a canonical surjective pairing ~/" G x k*--.HZ(G, k*). ([15]), proposit ion 9, p. 182 and , theorem 1, p. 222). Let n be a prime e lement of k, then Z ~ ~/(Z, n) gives an isomorphism of G onto HZ(G, k*) which is independent of the choice of n. Hence HZ(G, k*) is cyclic of o rde r n. Let a s G be the Frobenius au tomorphism and 7,F S G such that Zv(a) = = e 2~i/", then ~/(~i', n) is a canonical generator for HZ(G, k*) which does not depend on the choice of n. As in the introduction, let y deno te the image of fl in H2(G, k*). Then ~ = rl(Xr, n') for a unique t, t < t <_ n - 1. Let 1([1) be the ideal of r [ X o , X 1 . . . . , X , ] generated by

g(Xo,Xl , . . . ,x . )= 1-I (Xo +o'(Ox, +... + O ~ i < _ n - I

+ oi( ( , - t ) X , _ 1) - n'X'~.

Let (r, (, n') = r [Xo, .. . , X,]/I([1).

Proposition 1. (r, ~, n') is an integral domain.

Proof Since r [ X o, . . . , X,] is factorial, 1([t) is a prime ideal if and only if 9(Xo, X 1, . . . , X , ) is irreducible in k[Xo, X 1 . . . . , X , ] and the greatest c o m m o n divisor of the coefficients of g is one. The latter condi t ion is clearly satisfed. Suppose that gl , g2 ~ r [Xo , X ~ . . . . . X , ] are non-cons tan t polynomials such that ~/192 = g. gl and g2 are necessarily homogeneous gl (Xo . . . . , X . _ I , 0 ) . g2(Xo . . . . . X , _ I , 0 ) = g ( X o . . . . , X , _ I , 0 ) , but it is well known that g(X o, . . . , X , _ I , 0 ) is irreducible (cf. [4], t heo rem 2, p. 80). So we may assume that gi(Xo, . . . , X , _ 1,0) = a ~ r, a # O. Since gl is non-constant and homogeneous, 91 = aXi, for some j , j > I, but X. X g so certainly X, X g~.

Corollary 1. (r, 7,, n') is a .flat r-module. Proof By the proposition (r, ~, n') is a torsion free r-module. Since r is a principal ideal domain, (r, ~, n ~) is a fiat r - module (AC, I, ~ 2.4, proposition 3).

Let (r, ~, n ~) -denote the quasi-coherent graded ~2.~p~ ,-algebra defined by (r, ~, n'). Since r is noetherian, (r, ." ' ~" -. ~, n ) Is a coherent graded osp,,r gebra (EGA I, 1.5.1). By the preceeding corollary (r, ~, rc t) "is a flat o~.p,,~,- module (EGA, IV, 2.1.1).


Corollary 2. Proj ((r, (,ltt)~) is an integral (i.e. reduced and irreducible) r-scheme. Proof. Proj ((r, (, ~t) ~) and Proj ((r, (, ~t)) are isomorphic as r-schemes (EGA, II, 3.1.3), so by EGA II, 3.1.14 it suffices to prove that (r, (, 7t t) is integral domain.

Definition. An r-scheme r-isomorphic to Proj ((r, (, ~zt) ~) will be called a canonical model for V(fl).

The generic fiber F of Proj ((r, (, ~t) ~) can be regarded as a k-scheme. Since Proj ((r, (, Itt)~) is r-isomorphic to Proj ((r, (, 7zt)), F is k-isomorphic

"to Proj ((r, (, ~t) | k) (EGA II, 2.8.9). Since k is flat over r, (r, (, ~t) | k is k-isomorphic to k[Xo ,X1 , ..., X,]/(g). Thus wc have.

Proposition 2. The generic fiber of a canonical model for V(fl) is a Chatelet model for V(fl).

The following proposition will be needed in w

Proposition 3. Let k be a p-field with ring of integers r and residue field ~:~. Let k, be the unramified extension of k of degree n, r, be the ring of integers in k,, and ( be a primitive q " - 1 root of unity in k,. Let ~ ~ Gal (k,/k) be the Frobenius automorphism and A( 0 = det (o'i((J)) o_<i.j<._ 1. Then r , = r [ ( ] and if n is odd, A(() is a unit in r. Proof. The first assertion is a well known fact ([11], proposition 16, p. 84). A(O is a Vandermondc determinant so

A(O = l-[ (ai(O - aJ(()) and t~(A(O) = 1--[ ( O'i+ 1(0 - aJ + 1(()). n-- l~ i>j~O i>j

For odd n multiplication by o" induces an even permutation on {ao, ~, ~2, . . . , t~,-1). So a(A(0) = A(0 a n d A(O ~ k. Since A(0 ~ r,, A(0 ~ k n r, = r. {(, o'(0, . . . ,an-l(O} belong to distinct cosets of (~) in r,([15], theorem 7, p. 16) so for i> j , a i ( ( ) - oJ(O is a unit in r,. Hence A(O is a unit in rn, therefore a unit in r.

We now define two types of models for Seved-Brauer schemes defined over arbitrary local fields, not necessarily complete. First we need some algebraic preliminaries.

Let A be a discrete valuation ring with group of units A* and quotient field K. Let L/K be a cyclic extension of degree n, n odd, and let B be the integral closure of A in L. Let G = Gal (L/K) and let T be a generator for G. For x ~ L define A(x)= det (zl(xJ))o~l, js .. The argument used in the proof of proposition 3 shows that A(x) ~ K. Let Ja/~ denote the dis- cdminant of B with respect to A.

30 Stuart Turner

Proposition 4. Assumptions and notations being as above the fol lowing conditions are equivalent:

i) There exists z ~ B such that L = K(z) and A(z) ~ A*. ii) There exists z E B such that B = A[z] and 6B/A = A.

iii) There exists z 6 B such that B = A[z] and A(z)E A*.

Prooj~ i)implies ii). By [16], theorem 30, p. 307, A(z)Z ~ '~BtA" Since A(z)2 ~ A*, 6B/A = A. By the same theorem 6n/A = A(z)2A if and only if

{1, _,,~ ,2, ...,- '"-~ j~ is a basis for B as a free A-module. Hence B = A[z] . ii) implies iii). Since z ~ BI the minimal polynomial f ( X ) for z over K yields an equation of integral dependence f ( z ) = 0 for z over A ([-16], theorem 4, p. 260). B = A(z) implies L = K(z) so f ( X ) has degree n and { 1, z, ..., z"- 1 } are linearly independent over K. Therefore { 1, z,.. . , z"- 1 } is a basis for B as a free A-module. So 5B/A = A(z) zA- By assumption 5S/A = A SO A(z)Z ~ A*. Since A(z)~ K and A is a discrete valuation ring, h(z) ~ A. That iii) implies i) is obvious.

Remark. It follows from i) that any conjugate r~(z) of z also satisfies the conditions of the proposition.

Remark. The equivalent conditions of proposition 4 are satisfied in two important cases: a) A has perfect residue field, B is a discrete valuation ring, and mA, the

maximal ideal of A, is unramified in B ([1 I], proposition 12, p. 66). b) m A splits completely in B, i.e. there exist n distinct prime ideals of B

l y i n g . o v e r m a, and n < card (A/m,O ( [ I I ] , exercise 3, p. 67).

L e t / ( be the completion of K and let V be a Severi-Brauer K-scheme of dimension n - 1, n odd, such that V XspecKSpec g, is a trivial Severi- Brauer /('-scheme and such that V is split by L/K. Let z ~ B satisfy the

conditions of proposition 4 and let dp ~ A* ~ NL/g ( ~ ) for each complet ion of L such t h a t / ( = ~. Let A be the completion of A and I be the ideal

of f l [Xo , . . . , Xn] generated by I-I (xo "~ "~i(z).e~71 "~ "ci(z2)X2 --]- . . . + O ~ i ~ < . - i

+ Ti( zn- 1)X,, - 1) - ~ X; . Let (A, z, ~b) = A[Xo, ... , X,,]/I. The arguments used to prove proposition 1 and its corollaries show that (A, z, ~b) is an integral domain and a flat A-module. They also show that Proj ((A, z, (k)") is an integral A-scheme. Definition. An ,4-scheme A-isomorphic to Proj ((A, z, ~b) ~) will be called a completed model with respect to L / K for the Severi-Brauer K-scheme V.


Proposition 5. I f mA splits completely in B, then the graded 74-algebra (.4, z, d~) is isomorphic to 't[Yo, ~ , .--, Y,]/(Y0 Yt --- Y~- t - $ Y~)- Proof Since the minimal polynomial f ( X ) for z over K yields an equation of integral dependence f ( z ) = 0 for z over A, each of the conjugates ,/(z), i = 0, 1 , . . . , n - 1, of z in L is contained in B. On the other hand, f (X) decomposes into linear factors in /(, the completion of K, because m,t

�9 splits completely in B. Let z' e / ( be a root of f ( X ) = 0. There is a unique K-embedding 2 of L i n / ( such that 2(z) = z'. Let x e B, since x is integral over A, 2(x) is integral: over .,1, but A is integrally closed, so 2(x)~ .4; hence 2(B)~ A. Let zi(z ') denote 2(zi(z)), i = 0, 1 ... . , n - 1, zi(z ') ~ .4.

�9 (z') ~(z '2) . . . , ( z '" - ' ) Let D =

zn-~(Z, ) .cn-I(Z,2 ) . . . ~.n-l(Z,n-1)

\10 0 0 0 ~ /

Det D = A(z'). By assumption A(z')~A* ~ A*, so D ~ GL (n,~). Let Yi = (Xo + Ti(z')X1 + ... + zi(z '"-l)X,-x), for 0 _< i <_ n - 1

Y . = X . .

Then D defines a degree preserving ,,]-algebra isomorphism of A[Xo,.. . ,X~] onto t][Y o .... , Y~] which induces an isomorphism of .4[Xo, . . . ,X , ] / I onto A[Y0, ..., Y,]/(Yo Yx ... Y ~ - I - ~ YD, as required.

Corollary. -4lYe, .-., Y.]/(Yo YI... Y.- t - (a Y".) is a f iat A-module.

We conclude w with the definition of another type of model for a Severi-Brauer scheme defined over a local field. Let K, L, A, B, G and V be as above. Let x ~ B, such that K(x)= L and x is contained in all maximal ideals of B. Let (~mAc~N~, j~(~ for each completion of L such that g c ~. Let J be the ideal of A[Xo, . . . , X.] generated by

1--[ (Xo + ~(x)Xl' + Ti(x2)X~ + . . . + ~i(x"-l)X._l) - 4,XL O < i < n - - 1

Let (.4,x, c~)= A[Xo, ..., X J / J . As before we see that (,4,x, ~b) is an integral domain and a flat ,4-module and that Proj ((,4, x, ~) ) is anintegral A-scheme.

Definition. An A-scheme A-isomorphic to Proj ((A, x, ~)~) will be called a degenerate model for the Severi-Brauer K-scheme V.

32 Stuart Turner

w Specialization of models for Severi-Brauer schemes.

Let k be a p-field with ring of integers r and residue field F~. Let k. be the unramified extension of k of degree n, r. be the ring of integers of k., and p. be the maximal ideal of r.. Let p : r . - * r . / p . = F~. be the projection homomorphism; for x e r. , let 2 = p(x). p extends naturally to a homomorph ism r . [X o, ..., X.]--* Fo.[Xo,..., X. ] which will also be denoted by p. Let G = Gal(k.]k) and a e G be the Frobenius automorphism. Let G = Gal (F~./F~) and 6 e G be the Frobenius automorphism.

Lemma 1. Let S be a symmetric polynomial in 7/[X o . . . . . X . _ x ] with each coe.~cient ,equal to +_ 1. For all x ~ r., let s(x) = S(x, a(x), ~r2(x),-.., ~ - '(x)). Then S(x) = S(~, 7r(~) . . . . , ~ ' - '(2)) = s(2).

Proof Since each coefficient of S equals ++_ 1, s(x) = S(~, a(x), a - ~ ) , . . . , ..., o ~- r(__x)). Since a is characterized by the property a(x) = x ~ (rood p,), a~(x) = x ~ = 2q~= ff~(2), for all i, 0 <_ i <_ n - 1.

Corollary. I f x ~ G such that A(x) ~ r*, then { 1, t~i(~), ffi(22) . . . . . ffi(~.- 1)} is a basis for ~:q. over gzq for each i, 0 <_ i <_ n - 1.

Proof A(x) is a symmetric polynomial in {x, a(x), a2(x) . . . . . a"- l(x)} with each coefficient equal to _+ 1, so A(x) = A(2). A(2) e r* implies A(x) ~ g:~*, so A(~)= - i - det (o" (x~))o<_i.j. _< . - l # 0. Hence { l, ffi(2),.., t~i(~,- i)} is a basis for U:~. over Fq.

Let z e r . such that r . = r[z] and A(z)~r* . Let ~ r and M = r[Xo, ... X.] / I , where I is the ideal generated by f ( X o, X 1 . . . . , X . ) = [ I (X ~ + ai(z)X l + . . . a i (~- I)X._ l) - ~X.". Then Proj (M) is an

O_<i_<n--I

r-scheme and Proj (M)xs~r ~, can be regarded as an U:q-scherae. Let M = F~[Xo, ..., X . ] / ] where I is the ideal generated by

[ l (Xo + + ... o'(r-')x._,) -

O<_i<n--1

Proposition 1. Proj (M) Xs~ c, ~:q and Proj (M) are isomorphic ~:~-schemes.

Proof The closed fiber of Proj (M) is canonically isomorphic t o Proj (M | Fq) (EGA II, 2.8.9) so it. suffices to prove that M | ~:q and M are isomorphic graded Fq-algebras. Since M is a flat r-module (cf. corollary I of proposition l, w M | ~:q is isomorphic as an ~:q-vector space to Fq[Xo, . . . ,X . ] / I ' , where I' is the ideal generated by p ( f ( X o . . . . X.)). Write f ( X 0 , . . . , X.) = ~ S~(z, ~(z) . . . . . a"- l(z))X~) - ~bX~, where ct is a

I~1 =. multi-index and where the S~ are symmetric polynomials as in the lemma.

The zeta function ofa birational Severi-Brauer scheme 33

By the lemma pOC(Xo, ... X~)) = (I '~P(S~(z' a(z),.., on- l(z))X') - p(~)X~ =

= ( z s~(~, o(~), . . . ~ - ~(~)) - ~ x ~ =

= [ I (Xo + ~ ( ~ ) x , + . . . # ' ( ~ - ' ) x . _ , ) - ~ x ~ . O<_i<_n--1

Corollary. Proj (M ~) Xsp~, I=r and Proj (M ~) are isomorphic Fq,schemes. Proof. EGA II, 3.1.3.

Remark. The corollary applies in two important cases: first, if Proj (M ~) is a completed model with respect to k,Jk for "a trivial Severi-Brauer k- scheme and second, if Proj (M ~) is a canonical model for V~), a non- trivial SeveriTBrauer k-scbeme.

We now calculate the zeta function of P.roj (M) in case n is prime, n > 2. Recall that the zeta function ~(T) of an Fq-scheme X is the formal

powe C series in Y such that l og ( ( Y ) = ~ N o T~" where No denotes v = l I ) "

the number of [q~-valued points of X. There arc two cases to consider:

I) ~b ~ r*, so ~ ~ Fq*, and II) ~b r r*, so ~ = 0.

Case I) Let ~ bc an algebraic closure of F~. The Fq,-valued points of Proj (M) correspond to the Fq~-rational points on the variety W in P,(F-~) defined by

11 (Xo + ~'(~)xl + . . .~'(~'-l)x._,)- ~x~=o. O < i < n - - 1

Lemma 2. N l = q"- l + q~- 2 + ... + q + l. Proof. Since {1,~i(~), ... ~1(~,-1)} is a basis for Ft, over Fq, for each i, 0 < i < n - 1, Wc~ {X~ = O} contains no F,-rational points. The points of Wbclonging to the complement of the hyperplane X, = 0 can be identified with the points of the variety W' in A,(F,) defined by

[-[ (to + #'(~)rl + ... ~(r-')Y._ I) - ~ = 0. O~_i~_n--1

Since {1,~ , . . .~ '-1} is a basis for ~:r over Fq, (yo ,y l , . . . ,y~_x)~W' for y ie I:~ ff and only if N(y o + zy 1 - I - . . . ~ - t y n _ l ) - - - - ~ , where N denotes the norm, N:[:~.~--* F*. Since N is surjective, ker N contains q ~ - l + + q~- 2 + . . . + 1 elements. By assumption ~ ~ F* so there are ~ - 1 + . . . + + I points {Yo, -.., Y,- i) in A~(F~) satisfying N(yo + zyx +... + z'- ly._ I) = --~. Hence N I has the required value.

34 Stuart Turner

Lemma 3. Let v ~ N such that nJ( v, then No = qOtn- 1~ + q~tn-2 ) .~_ .....]_ qV + 1.

Proof. Since n is prime, ~=q~ n ~=qn = ~=q and ~=q~ and ~=q~ are linearly disjoint over ~=~. Hence {1, 81(~) . . . . ~i(y,-~)} is again a basis for ~ over ~=~ and the argument of lemma 2 shows that N~ has the required value.

Lemma 4. Let v ~ N such that n]v, then Nv=qntn-1) +qVtn-2) + ... +qV + 1. Proof. Since nlv, and t~i(~J)~ ~=q~ for all i,j, O<_i , j<_n-1 . By assumption A(z)~ r*, so A(~)~ U=*. Therefore (ai(~J))o<_i.j<n_ 1 ~GL(n, Fq~). Let, Y~ = X o + 0~(~)X1 +..~ + 0~(~n- x)X~_ I, 0 < i < n -- 1

Y, = X~.

The argument of,proposition 5 of w shows that W, considered as a variety defined over ~=qv, is Fq~-isomorphic to the projective variety IV" in Pn([=~) defined by YoYI ... Y ~ - I - ~ yn =0. Hence to calculate N~ it suflrmcs to count the ~=q~-rational points on W".

W"c~(Y~=0} can be regarded as a variety in Pn_l(~=q)..The affine cone over this variety is the variety in A n (U=q) defined by Y0 YI ... Yn- 1 = 0. This affine variety contains (q~ 1), 0=q~-rational points, so the

projective variety W "c~{ Yn = 0}. contains qno - (qo - 1)n - 1 q ~ 1 0=q~-rationa!

points. W"c~{Yn#0} can be identified with the points on the variety in An([:~) defined by Z 0 Z1... Z~_ 1 - ~ = 0. This variety contains (qo - 1)~- 1

F~-rational points. So N~ = q~t"-') + q~!~_z~ + ... + qO + 1. Combining temmas 2, 3, and 4 with proposition 1 and the remark

fol lowing the corollary of proposition 1 yields.

Proposition 2. Let n be a prime, n > 2, and let V be a trivial Severi-Brauer k-scheme of dimension n - 1. Then No, the number of ~:q~-valued points on the closed fiber of a completed model for V with respect to k,/k, is given by No=q~tn-') +qd~-2) +. . .q~ + 1, for all v ~ N .

Corollary. The zeta function of the closed fiber of a completed model for V is given by

~(T) = [(1 - qn-ZT)(1 - qn-ZT). . . (1 - q T)(1 - T)] -1.

Case II). Since 4~ - 0 , the ~:,v-valued points of Proj (/q) correspond to the F,.-rational points on the projective variety W 1 in P,(~:q) defined by 0-< ~[~- z (X~ + ~(~')X1 +"" a2(y'-l)X~- 1) = 0. Observe that the point

(0, 0 , . . . , 0, x~), x, 4= 0 is an Fq-rational point, hence No k 1 for all v s N.

The zeta function o f a birational Severi-Brauer scheme 35

Lemma 5. Let v~ N such that nJ(v, then N~= 1.

Proof. Since n2(v, {1,~i(~) . . . . ,0i(~'-~)} form a basis for ~:a,~ over ~:~, for each i, 0 _< i < n - 1. Hence the only ~:~-rational point on I4/1 is (0,0 . . . . , x~) ,x ,~O.

Lemma 6. Let v~ N such than n I v, then N v = n(q ~ .... ~ + qOt.- 2~ + . . . + qO) + I.

Proof Since n[v, ~:qnc ~:qo and #i(~J)s D:q. for all i , j , O < i , j < n - 1 . By assumption A(z) ~ r*, so A(~) ~ D:*. Therefore (#i(~j)) ~ GL(n, ~:~0. Let Y~ = X o + O~(~)Xt + ... + ~i(~,- t)X~_ t, 0 -< i <_ n - I. The argument of proposition 5, w shows that I4,'i, considered a variety defined over Fq~, is isomorphic to the project ive variety W~ in P,(~:q) defined by Yo YI--- Y~- t"-- 0. W t' is a union of the hyperplanes { Y/= 0}, i = 0,... n - 1. Their common intersection is the point (0, 0 .... ,0, y,), y, # 0. The com-

plement of this point in each hyperplane contains qV(-:, + qV~--2~... + q~ 0:~-rational points. Counting the points with their appropriate multiplicity gives N~ - 1 = n(q~ + qOt"-2~ + ... + qO). Combining lemmas 5 and 6 with proposition 1 and the remark following the corollary of proposition 1 yields.

Proposition 3. Let n be a prime, n > 2, and let V(fl) be a non-trivial Severi- Brauer k-scheme of dimension n - 1. Then No, the number o f ~:q~-valued points on the closed fiber of a canonical model for V, is given by Nv = I,

i f n)(v, and N~=n(q~ +q~ ... +q~)+ 1, if nlv.

Corollary. The zeta function o f the closed fiber o f a canonical model for V is given by

((T) --- [(1 - (q"-~ T)") (1 - (q,,-2 T)")... (1 - (q T)") (1 - T)] -~.

This concludes the discussion of Case II).

Let A be a discrete valuation ring with maximal ideal mA and quotient field K. Let L/K be a cyclic extension of degree n, n odd, and let B be the integra! closure of A in L. Assume that m,~ splits completely in B and that n < card (A/mA). L e t / ( be the completion of K and let V be a Severi- Brauer K-scheme of dimension n - 1, n odd, Such that Vxspec K Spec I( is a trivial Severi-Brauer/(-scheme and such that V is split by L/K. By proposition 5, w a completed model for V with respect to L / K is /i- isomorphic to Proj (N) where N = / ] [ Y o, Y1,.-., Y,]/(Yo Y1-.-Y,-t - qb Y~) with ~b ~/i*. Let F = ; t /mj4 and let q~ be the image of ~b in F. The closed fiber of Proj (N) is canonically isomorphic to Proj (N | (EGA II,

36 Stuar t Turner

2.8.9). Since N is a flat A-module (corollary of proposition 5, w N | is isomorphic as an F-vector space to

F[Yo, . . . , Y 3 / ( r . . . - Y : ) .

Assume now that F = ~:~, then the ~:~-valued points on the closed fiber of Proj (N) correspond to the Fq~-rational points on the variety

W"in P.(~q) defined by Yo I"1---Y.-1 - ~ Y: = 0. The argument of l e m m a 4 shows that No = q~'- ') + ... + q~ + 1.

Proposition 4. Let n be odd and let V be a Severi-Brauer K-scheme of dimension n - 1, n odd, such that V x s~rSpec I( is a trivial Severi-Brauer I(-scheme and such that V is split by L/K. Then N~, the number o f ~=q~- valued points on' the closed fiber of a completed model for V with respec t to L[K, is given by Nv=q~ +. . . +q~ + 1, for all yEN.

Corollary, The zeta function of the closed fiber of a completed model for V with [espect to L/K is given by

~(T) = [(l - q ~ - ' T)(1 _ q , - Z T).. .(1 - q T)(1 - T ) ] - ' .

Let K, L, A, B, F and V be as above. Let G = Gal (L/K) a n d let z ~ G be a generator for G. Let x ~ B, such that K(x) = L and x is conta ined in all maximal ideals of B. Recall that a degenerate model for V is ,4-isomorphic to Proj (P) where P = A[Xo, ... X~/J , J being the ideal generated by 1-[ (Xo+z~(x)X1+.. .+zi(x~-~)Xn_l)-dpX~, with ~b~mA. As

O<i<a--1

before, the closed fiber of Proj (P) is canonically isomorphic to Proj (P| Since P is a flat ,4-module, x belongs to each maximal ideal of B, and ~b~m~, P | is isomorphic as an F-vector space to F[Xo,. . . , XJ /X~. Assume now that F = Fq. Then the F~u-valued poin ts on the closed fiber of Proj (P) correspond to the Fq,-rational po in t s on the variety in P,(U=q) defined by X~ = 0. This is a hyperplane, so we have.

Proposition 5. Let n be odd and let V be a Severi-Brauer K-scheme of dimension n - 1, n odd, such .that V Xsp~r Spec K is a trivial Severi-Brauer l(-scheme and such that V is split by L/K. Then No, the number o f ~=~,- valued points on the closed fiber of a degenerate model for V, is given by No = q~n-1) + . . . + qO + 1.

Corollary. The zeta function of the closed fiber of a degenerate model for V is given by

~(T) = [ ( l - r T) . . . (1 - q T)( l - T ) ] - ' .

The zeta function o f a bitational Severi-Brauer scheme 37

~ . Cydic Brauer classes. Throughout the rest of the paper k will denote an algebraic number

field. Let [1 be the set of places of k; for v ~ [1, let ks be the completion of k at v. Let Br(kv) be the Brauer group of ks. In case k s is isomorphic to R, Br(ks)-~ {_+ 1}; in case ko is isomorphic to C, Br(k,)~_ {1}; and in case ks is a p-field, Br(kv) is canonically isomorphic to the group of roots of unity in C*. For each v there is a canonical map Br(k) ~ Br(k~). These maps give rise to an injection Br(k)-* l-lBr(ks)([3"], theorem 2,

ve[~

p. 16; [15], p. 252). Let y ~ Br(k) and let ~v be the ' image of 7 in Br(k~). Let n be the order of 7; n is clearly equal to the least common multiple of the orders of the ~ .

Let S = 1}, S is finite ([15], theorem 1, p. 202). In case o e S is a finite place, l~t. kv,,~ be the maximal abelian extension of k~. (in a fixed algebraic closure of k~). Let ~ = Gal (k,,ab/k s) and ~ , be the open subgroup of r with fixed field ks.,, the unramified extension of k of degree in. Let ~b: t ~ - * ~ J R . be the projection and o s ~ ~ . /R .be the Fro- benius automorphism. Let ~k s be the character on ~,/R, such that ~#~(o~) = = e 2"~/'. Let as. k~ --* ~ be the canonical morphism of local class field theory. Then X~ = ~ks o ~b o a s is a character of order n on k*. In case v e S is an infinite place, let ~ be the non-trivial character on ~v = Gal (C/R). Let ao: R* ~ ~o be the canonical morphism. ~ = d/s o a,, is the character on R* which maps R*+ to I and R* to-I.

Let P(n, S) be the group of elements z e k* such that z e(k~)" for

all vr Assume that P(n, S) = (k*)" in order to avoid the notorious "special case" discussed in [3], pp. 93-6. Let Jk denote the idclc group of k. There exists a chacter X of Jk of ordcr n, trivial on k*, such that the restriction ofx to k~* is equal to ~ for all v e S ([3], theorem 5, p. 103). Let kob be the maximal abelian extension of k and ~= Gal(k,a,/k) and a: Jk~ be the canonical morphism of ~obal class field theory, a is surjectivekThe map induced by a from ~ , the group of characters of a, to Jk �9 the group of characters of J~, gives an isomorphism of ~ ^ onto the group of characters of Jk of finite order, trivial on k*([15], thcorcm 5, p. 271). So Z can be regarded as a character on ~. Let k' be the fixed field of ker X. By construction k'/k is a cyclic extension of degree n such that for all v~S,k' | is a field, k~; in case v is finite k:,. is ko-isomorphic to ks.,; in case v is infinite, k:, is ko-isomorphic to C. For every place v e fl the degree of k~ over ks is a multiple of the order of ~o in Br(ko), so ~ is a cyclic Brauer class of degree n with splitting field k' of degree n over k ([15], proposit ion 5, p. 253).

l e a n be regarded as a character on ~ = Gal(k~,~Jk) constant on cosets of 6[ in ~ and l s can be regarded as a character on ~s = Gal(k~. ~,~/k~)

38 Stuar t Turner

constant on cosets of 0( o in ~o. Let rl:~ x k * ~ Br(k) be the canonica l pairing for k ([15], p. 181) and go: ~v x k* --. Br(ko) be the canonical pair ing for k o. There exists 0 e'k* such that y = r/(Z, 0). 0 is defined up to an e lement of Nk,/k(k') and ~o = r/o(Zo, 0). Let v e S, v finite, and rc be a pr ime e lement of ko. F r o m w recall that there exists a unique integer t~, 1 < to <- n - 1, such that Yo = r/o(X, n'v) �9

Proposition 1. Let A denote the ring of integers of k and let T be a finite set of places of k such that S r~ T = c~. Then there exists 0' ~ A satisfying the conditions:

i) ~ = r/(X, 0'), ii) for all v~S, v finite, ordo(O' ) = to,

iii) for all v ~ ~ v finite, ordo(O') >_ 1.

Proof. Let 0 be as above and let r* be the units in the ring of integers of ko. FoL v e S , v finite, Yo = r/o(Zo, 0) = ~/o(Zo, rvv) so 0-1 n,~ e Nk~,/k ~ (k'~). Since k'~/k o is unramified of degree n, Nk'~/k~ ( k ' ) = {Tz ~, r*}a~z ([15], p. 139 and corollary p. 226), so t o - ordo(.O)= bon for some bo~Z.

Fo r x e k ' , Nk,/R(X)=Nk~/kv(X) ([15], corollary 3, p. 58) a n d ordo (Nk~,/kv (X))-~-n ordw(x). Hence if ord~(x)= bo, then ordo(O Nk,/k(X))= t o. For any v ~ f~, v finite, if x ~ k' has sufficiently high order at a place w of k', w[ v, then "ordo(O Nk,lk (X)) >__ I.

Let U = { v ~ f ~ [ v q ~ S u T a n d ordo(O)<O}. U is a finite set. L e t f~k, be the set of places of k'. Using the strong approximat ion t h e o r e m ([5], p. 67) in k' select y e k' such that

a) for v ~ S, v finite, and for w ~ ilk', W I V, ord~(y)= bo, b) for v ~ T, o finite, and for some w ~ f~k,, w l v, ordw(y) is sufficiently large

that ordo(O Nk,/k (y)) >__ 1, C) for o ~ U and for some w ~ f~k,, w l o, ord~(y) is sufficiently la rge that

ordo(O Nk,/k (y)) >__ O, d) for all finite v, v ~ S u ' T u U and for all w ~ f~k', W[ V, ord~(y)> O.

Then for v e S, v finite, ordo(O Nk'/k (Y))= to; for v ~ T, v finite, ordo(O N k,/k (y)) > 1 ; and for all other v ~ f~, v [mite, ordo(O Nk,/k (Y)) > 0. So 0' = 0 Nk,/k (y) ~ A. 0' satisfies i ) - iii).

Remark. Proposit ion 1 was proven under the assumption that k, S and n do not give rise to the "special case". Proposi t ion l is sufficient for applications later in this paper. However, a stronger result can b e established using a similar argument. We describe the result as it m a y be of independent interest. Again, let k be an algebraic number field a n d let

~ Br(k). ~ defines a central division algebra over k, O(y). The index i = i(~)


of D(y) is defined by [D(y): k] = i(~) 2. The Albert-Brauer-Hasse-Noether theorem ([7], cf. [3] corollary, p. 105) implies that y is a cyclic Brauer class of degree i (y)- even under the conditions of the special case. Let S ' = {v~f~lyo~ 1} and let S O be the subset o f f l defined on p. 95 of [3]. For all v ~ S', v r S o, let Rv, i denote the subgroup of d o = Gal (kv.,b/ko) with fixed field ko. ~. Let a o ~ 6~o/Ro, ~ be the Frobenius automorphism and Go be the character on ~o/Rv.i such that ~o(o'o)= e 2"~/~{y~. Let Zo be the character of order i on k* defined by ~k o as above. Then there exists a character ~ of Jk of order i, trivial on k*, such that the restriction of Z to k* equals Zo([3], theorem 5, p. 105). For v E S', v r S o, let t o be defined as above. An argument similar to the one used in the proof of proposition 1 gives.

Proposition 1'. Let A denote the rin9 of integers of k and let T be a finite set of places such that S ' n T = qb. Then there exists 0'~ A satisfying the following conditions:

i) ~; = r/(X, 0') ii) for all v ~ S', v finite, v r So, ordo(O') = to

iii) for all v ~ T, v finite, ordo(O') > 1.

Before giving an important coroilary to proposition 1 we recall a result of Hensel concerning the discriminant of a finite extension of an algebraic number field. As before, let k be an algebraic number field and A be the ring of integers of k. For v ~ f~, let Po denote the prime ideal of A defined by ~. Let L/k be a finite Galois extension of k of degree m with G = Gal(L/k). Denote the elements of G by r~, 0 < i <_ m - 1. Let B be the integral closure of A in L and let 6n/A be the discriminant of B with respect to A. For z ~ L , let A(z)=det(~,(zJ))o_<i._<~_l; A(z)q:0 if and only if L = k(z). {A(z)2lz ~ B, k(z)= L} generate an ideal ln/a of A.

= if pol and p~Xfn/a, then Po is called a common inessential factor of the discriminant of L/k ([8], p. 439). From Hensel's criterion ([8], p. 442) it follows that if p~B is a prime ideal of B, then Po is not a common inessential factor of the diseriminant of L/k. A necessary condition for Po to be a common inessential factor of the discriminant is that card (A/po) < th. This condition is also sufficient if po splits completely in B. We apply these results in case L = k', the extension of k described above, under the additional assumption that n is and odd prime.

Corollary. Notations beincd as in Proposition 1, assume that [k': k] = n is and odd prime. Let Tl={v~f~lp~lln/a}. Then there exists O'~A sa- tisfyinff the following conditions:

i) r = r/(X, 0'),

40 Stuart Turner

ii) for all v ~ S, v finite, i ord~(O') = t~, iii) for all v ~ T, v finite, ord~(O') > 1.

Proof As n is odd, k, rg and S do not give rise to the "special case". As n is prime, po [In/a implies that po is totally ramified in B or that po splits " completely in B and card (A/po)< rn. Hence S c~ T 1 = ~ . The corollary now follows from the proposition.

Throughout the rest of ~4 the assumptions will be those of the corollary. We let 0(instead of 0') denote an element of A satisfying con- ditons i ) - iii) of the corollary. Define

T 2 = {v E nl oes T,, ord~(O) >_ 1}, so S u T~ u T 2 = {o nlorao(o) >_ ~ }.

For any v such that k',/ko is unramified of degree n, the restriction m a p Gal (k',/kv)--*Gal(k'/k) is an isomorphism. Identify these two groups using this isomorphism. Then A(x) is well-defined for any x ~ k~,: A: k , kv is a polynomial map, therefore continuous. Let q~ be the cardinality o f the residue field of k~. k" is generated over k~ by (w, a primitive q ~ - 1 root of unity. Recall that A((~)e A*, the group of units of A o (proposition 3, w

Lemma. Let B , be the ring o f integers o f k',. 7here exists e > 0 such that for all z ~ k' , lz - (~,l w < e, the following conditions are satisfied:

i) k'~(z)= kv((.)= k'~, ii) z ~ B . ,

iii) A(z) A*.

ProofLet ,~l = rain I ~ ( , - - ( , , ] , . L e t z e k ' s u c h t h a t [ z _ ( , l , < - s t . r ~ id

By Krasner's lemma ([9], proposition 3, p. 43) kv((. ) ~- kv(z), so k~, = k=(z). Since ( . e B . , for some e 2 > 0, [ z - ( . [ . < e 2 implies z ~ B. . Since A: kL -* --* ko is continuous and A((.)~ A*, for some 8 a > 0,1 z - ( - I - < % implies A(z)~A*. Set e = m i n ~ , i = 1,2,3.

Proposition 2. Let B be the ring o f integers o f k'. 7here exists z t ~ B such that

i) k(zt) = k', ii) for all v ~ S, v finite, A(zl) ~ A~*

iii) f o r all w]v, v~ T i u Tz, ord~,(zt)>_ l.

Proof If there is no finite v ~ S and T 1 u T 2 = O, then ii) and iii) are vacuous and i) is obviously satisfied for some z t ~ B. If there is no finite v E $, but T 1 w T 2 # O, let z E B be a primitive element for k' and let a ~ A


be such that ord~(az~>l for all w [ v , v ~ T l u T ~. Then z x = a z satisfies i) and iii), while ii) is vacuous. Finally, assume that there is a finite v~ S. For each such v select 8o > 0 as in the lemma; let 8 = mine r Using the

mS

strong approximation theorem in k' select z I e k' such that [z I - (w] ~ < for all w[v, v ~ S , v finite, such that ord~(zl)> 1 for all w[v, v E T 1 u T z , and such that ord~(zl) > 0 for all other finite places w of k'. zt ~ B and ii) and iii) are satisfied. To verify i) observe that [-k(zl): k] _> [k(Zl)~: k J > n for all finite v ~ S, so k(zl) = k'.

By definition of T 1 , for each finite v ~ fl, v ~ T1, there exists an x~ ~ B such that k ' = k(x~) and A(x~)eeA * (or, equivalently, A(xo)~A*). Let ~1 = { v ~ l v finite, v r 2, A(zl)~A* }. Z t is finite set. Let v o be a finite place of k, voq~SoT t u T 2 u Y ~ , such that there is only one place w o of k' lying over Vo, so k'o]k~o is unramified of degree n. Let q~o be the cardinality of the residue field of k~ o and (wo be a primitive ~o - 1 root of unity in k~o.

Proposition 3. There exists z 2 ~ B such that

i) k(z2) = k', ii) for all v~ Y~l, A(z2)eA*.

Proof. I f Z 1 = O, ii) is vacuous and i) is obviously satisfied for some z 2 e B. The rest of the proof being essentially the same as the proof of proposition 2 it suffices to remark that one choose z 2 e B close to (wo in the topology of k~o and close to each x~ in the topology of each k'~,w[v,v~Y_, I.

w A model for Severi-Brauer schemes over algebraic number fields.

Let Y = Spec A and for v~D., v, finite, let Ao denote the valuation ring in ko. For an A-scheme X, let X~ denote the Ao-scheme X x r Spec A~. Let X o denote X x r Speck. Let V be a Severi-Brauer k-scheme of dimension n - 1, n an odd prime, defined by y~Br(k). Let ~v be the image of y in Br(kv). For each ve i l , let oV= Vxsp~kSpecko . The purpose of this w is to prove:

Theorem 1. There exists a quasi-coherent graded O_r-algebra 6e, f iat over Or, such that Proj (6 ~) is a projective Y-scheme and such that

i) Proj (6e) o is a Chatelet model for E ii) For all v~S, v finite, Proj (6e)o is canonical model for I/(Yv).

iii) For all v ~ T l u T 2, Proj (6e)v is a degenerate model for~ V. iv) For all other v ~ O~ v finite, Proj (,Se)o is a completed model for oE

42 Stuart Turner

The ctmgtruction of 50.

We make the following notational convention: if D is an integral domain with quotient field E, F is a finite Galois extension of E with H = Gal (F/E), x ~ F such that F = E(x), q5 ~ E*, then the g raded D- algebra

D [ X o , X , . . . . , X,]/(1-I (Xo + a(x)X1 + ... a(x"- l )X,-1 - r X D tIEH

will be denoted by (D, x:, ~b). The quasi-coherent or-algebra 5 r is constructed by first assigning

to each member of a cover for Y, consisting of subsets of the form D(f) , f ~ A, a quasi-c0herent graded __or-algebra and then joining these sheaves using recollement de faisceaux (EGA 0, 3.3).

Let f~sl, be the set of finite places of k and let U 1 = f~si, - - ~~1 .

( S u T 1 u T 2 ) c ~ E I = ~ , so S u T 1 u T 2 ~ U t. Identify U 1 with an open subset of Yand let {Wil Wi c Ul}i~ t be a covering of U 1 by distinguished affine open subsets of Y. Each W~ = D(fi) for some f~ E A. The affine scheme (W~,s w) is canonically isomorphic to Spec As~. (As~ , zt, 0) is a graded AsTalgebra of finite type. (~As~, zt, 0) can also be regarded as a graded A-algebra. Let (As~,zt,O) be the associated quasi-coherent graded or-algebra. Then (Asi, z~, 0) ] W t is canonically isomorphic to the quasi- coherent graded OspecAsTalgebra associated to the Asralgebra (Asi , zt, 0) (EGA I, 1.3.6). For h, i ~ I, (As,, zl, 0 ) I w,~ wi and (AIi, zx, 0) [ w , , wi are isomorphic as quasi-coherent graded or-algebras because each is iso-" morphic to (As~r~, z I, O)~lvc,,,w~.

Let E a = {v ~ f~l v, finite, v ~ S u T t u T 2 or A(zz) r A*}. By construct ion Ea n E z = ~b. Let U 2 = f~I~,- Za" Identify U z with an open subset of Y. Let {Xj IXj = U2}~ s be a covering of Uz by distinguished affine open subsets of Y. If Xj = D(gj) and Xk = D(Ok), O j, 9k ~ A, then as before (Agr z2, O)~lxj,, x~ and (Ao~, z z, O) ~ I xj,,x~ are isomorphic as quasi- coherent graded or-algebras.

{W~, X~}~j.j. s is an o l i n cover for Y. In order to apply recollement de faisceaux to establish the existence and uniqueness of a quasi-coherent graded or-algebra 5 e such that 5elw i is isomorphic to (Ay~,za, O)~lw~ for all i ~ 1 and 5 a [ x~ is isomorphic to (A~, z z, 0) ~ I x, for all j ~ d it suttices to verify that (Afi , Zl, 0)~[lu Xj and (Agj, Z2, 0)~l Wir~X.~ are isomorphic. (Aft, z l , O) ~ [w~ x~ is canonically isomorphic to the quasi-coherent graded Osp,,af~ -algebra associated to (A f ~g~, zx , O) and (Ag~, z2, O)~ [ ve~x, is canonically isomorphic to t h e Os ,~ar associated to (A , z 2, 0). Since the functor ~ from the category of graded A~Ta lgebras to


the category of quasi-coherent graded Osp~a~:algebras is faithful, it suffices to prove

Proposition 1. (A f ig~, z l , O) and ( a f lej , z2, 0) are isomorphic 9raded A f ia~-

aloebras. Proof Let C be the Af~a-algebra A f ~j[ Yo, ..., Y,]/( Yo Ya ... Y,- t - 0 Y~). There is a A:~:algebra ~homomorphism of (Af t , , Zx, O) onto C which maps Xo+a' (zOX~+. . .+a ' ( z~"-~)X,_x to Y~ for i = 0 , . . . n - I and maps X, to Y,. This homomorphism is defined by the matrix

( 0 1 i) 1

, 1 .

Z l ~ �9 �9 �9

1 o~-l(z 0 ~"-'(z 2) a"-'(~-') 0 0 0

Similarly, there is an A:~9:algebra homomorphism of (A:~g~, z 2, O) onto C defined by the matrix

1 ~r(z2) cr(z~) ... a(~z -x)

0 0 0 ... 0 1 /

An easy calcuiation recall that n is odd) shows that a(Z~ l Z1) = Z~ 1 Z1, so Z~-I Z t has entries in k. By construction A(zh)= det Zh~ A* for all ve $ ~ n X j , h = 1,2. Hence Z~ t Z x has entries in A o n k and det Z~ 1Z x eA*c~k for all v e $ ~ n X i, so Z~l Z l e G L ( n , Avnk). Since A:~ j=

= (-] (A,c~k),Z~ x Z 1 has entries in A:~oj. Since det Z~ "t Z 1 is a unit I ~ W i~ X j

in each A o n k, it is a unit in Af~j. Hence Z21 Z 1 E GL (n, Af.~). So Z21Z1 defines an isomorphism of (A fig j, z I , O) and (A f ~ , Ze, O) as graded A f ~gj- algebras.

The properties of 6 a

The assertions of theorem 1 are proven in proposition 2-10.

Proposition 2. 6 a is a coherent 9raded o_:alaebra. Proof. By construction 5" is a quasi-coherent graded or-algebra of finite type, so since A is noetherian, 5: is a coherent or-module (EGA I, 1.5.1).

44 Stuart Turner

Proposition 3. Ae is a flat or-module.

Proof. Since Aa]w~ is isomorphic to (Ay~,z1,0)~[w~ and (Afi, zl , O) is an integral domain (cf. proposition 1,w 6 e is an integral domahn for all p e W~. Applying the same argument to 6el x~ we conclude tha t .Sap is an integral domain for all p e Y But for each p e Y,_op. r is either a discrete valuation ring or a field. So 6ep is a fiat op.r-module for all p e

Proposition 4. Proj (Ae) is a scheme of finite type over Y and the morphism f : Proj (A a) ~ Y is projective. Proof. Let 6e~ denote the homogeneous component of 6 e of degree i. By construction 6~ o is isomorphic to _o r, 3e~ is an 6eo-module of finite type, and 6a is generated by 6e~ (EGA II, 3.1.9). The two assertions now follow from EGA II, 3.4.1 and 5.5.1-2 respectively.

Proposition 5. Proj (6e) is integral i.e. reduced and irreducible. Proof. InTthe proof of proposition 3 we observed that S/' is integral (EGA II, 3.1.12) and that Y is an integral scheme. Since 6e o is isomorphic to o r, the conclusion follows from EGA II, 3.1.14.

Proposition 6. Let ~bo: Spec k ~ Y denote the canonical map of S p e c k to the generic point of Y Then Proj (A e) x r Spec k is a Chatelet model for the Severi-Brauer k-scheme V.

Proof. As before, let D(f) be a basic open set of Y contained in U 1 �9 The schemes (D(f), _oylo~I~) and Spec A I are canonically isomorphic. Identify them using this isomorphism. The inclusion homomorphisms A ~ / I f ~ k induce maps Spec k~Spec A I ~Spe c A; ~b o is the composite of these maps. Proj (6e) xr Speck and (Proj (Se)xr Spec AI)x s~cal S p e c k are canonically isomorphic as k-schemes (EGA 0, 1.3.2), so it suffices to show that latter is a Chatelet. model for V.

By the definition of Proj (S~) (EGA II, 3.1.2-3) Proj (S~)x r Spec AI is canonically isomorphic as an Al-scheme to Proj (F(D(f), 6a)). By construction Se [ DCf) is isomorphic to the quasi-coherent graded Os~ ~ a-'algebra (A I, z 1 , 0)~[D(/), so Proj (oCe)xr Spec A I and eroj ((A I, zl, 0) ~ ID(I)) are canonically isomorphic as A/-schemes. From the proof of EGA I I , 3.5.3 one sees that Proj((A f, z t, 0) ~ ID~)x s~r at Speck and Proj((AI, z l, O) ~ a l k ) ~ are canonically isomorphic as k-schemes. Since k is a flat AI-module, (AI, zt , 0) | k and (k, zt, 0) are canonically isomorphic graded k-algebras. So Proj((Ai, zl, O)~locf))Xs~c at Speck and Proj(k, zl, 0)~ a re canonically isomorphic k-schemes. The latter is a Chatelet model for V.

In order to state and prove the next proposition without introducing an awkwardly complicated notation we make the following convention:


(Ao, zi, 0) ~ (resp. (Ay, z i, 0) ~) will denote the gradedos~Aj(resp, os~r ) algebra (instead of the graded _Os~r algebra) associated to (Ao, zi, 0) (resp. (As, z i, 0)).

Proposition 7." Let v be a finite place of k, k v be the completion of k at v, A~ the valuation ring in ko, and ~/o: Spec Ao-" Y be the map induced by the inclusion A-*Ao. I f v e Ui, i= 1,2, then Proj(Se)Xr Spec A~ and Proj(A v, zi, O)~ are canonically isomorphic Ao-schemes. Proof. Assume v E U 1 and let D(f) be a basic set of Y contained in U 1 such that v ~ D(.f). As in the proof of proposition 6 one sees that Proj(ff) xr Spec A v and (proj(SP) xr Spec A f) x sp~.41 are canonically isomorphic as Ao-schemes; and also that Proj(5~')xrSpec A f and Proj (Aj.,z~, 0) are canonically isomorphic as Afschemes. From the proof of EGA II 3.5.3 one sees that Proj(Af, z l ,O)Xs~_atSpec Ao and Proj (AI, zi, 0)'| Ao)~ are canonically isomorphic as Ao-SChemes. Since Ao is~0btained from A s by10calization and completion, Ao is a flat A/- module, hence (A:, z l, 0)| Ao and (A o, z 1 , 0) are isomorphic as graded A~:algebras. So Proj(Ay, z l, O) x s~.~ AI Spec Ao and Proj(Av, z 1 , 0) are canonically isomorphic Ao-schemes.

Proposition 8. For all v~S, v finite, Proj(~f)x r Spec Ao is a canonical model fo r V(~). Proof Let q~ be the cardinality of the residue field of Ao and ~ be a primitive ~ - 1 root of unity in ko.,, the unramified extension of k~ of degree n. Let r be the Frobenius automorphism of ko., over ko. Since v ~ S, v e U 1 . By proposition 7 it suffices to show that (Ao, zl, O) is isomorphic as a graded Ao-algebra to an (Ao, ~, ~b) where ~b ~ Ao and ordo(O)= t, t being the positive integer defined in the first paragraph of w But by the corollary to proposition 1, ~4, ordo(O)= t. An isomorphism between (Ao, z l, O) and (Ao, ~, 0) can be constructed by noting that

2 .... ! / - 1 / ! " I ~(0 ~(~2) ... ~(~.- ,) ~ ( z x ) ~ ( - 1 )

r . . . . , ; ~-I(0 ... o"-I(~ ~-I) o~-l(zl) ... 0 o o . . . o o . . . o

r GL(d, Ao) (because A(~)~A~* by proposition 3, w and A(z~A*~ by proposition 2, ~4) and employing the argument used in the proof of pro- l~sition I.

46 Stuart Turner

Proposition 9. For all v~ T 1 ,-)7"2, Proj(6f)x r Spec Ao is a degenerate model for oE

Proof. Since v~ T~ w 7"2, v ~ U t . Let D(f) be a basic open set of Y contained in U~ such that v ~ D(f). The inclusion homomorphisms A ~ A/---, --. Aoc~k--, Av induce maps Spec Ao--* Spec Aoc~k ~ Spec A / ~ Spec A. Let Z~ denote the Aonk-scheme (Proj (6e) xy Spec A/) x Sp~a~Spec (A~c~k). Then Z~ Xsp~r ~ Spec Ao is A~-isomorphic to Proj(6e)xr Spec Av. As in the proofs of proposition 6 and 7 one sees that

ZI Xsp~,ta~,~k~ X Spec A~ and Proj((Ao n k, Zz , O)(~Au,'3k A~) ~

are isomorphic A~-schemes. Since Ao is a f ia t A v c ~ k - m o d u l e , (Aoc~ n k , z~,O)| and (Ao, z~,O) are isomorphic graded Ao-algebras. It suffices to show that k ' = k(z~), ord~(z 0 > 1 and that 0 s m,~c,k ~ Nk'/k~ (k~) for all places w of k', w] v. That the first two conditions are satisfied follows from proposition 2, N; that the last condition is satisfied follows from the corollary to proposition 1, N, and the definition of T 2.

Proposition 10. For all v ~ f2, v finite, v q~ S u T 1 u T 2, Proj(,9 ~) Xy Spec A v is a completed model for oV.

Proof. In case v~ U 1 the proof follows that of proposition 9 up to the final two sentences. Beyond this point to conclude the proof of proposition 10 for v~U 1 it suffices to show that k'=k(zl) , A(zl)~(A~c~k)* and that 0 ~ (A ~ c~ k)*, 0 ~ Nk:/k ~ (k') for all places w of k', w I v. By proposition 2, ~4, and the definition of Ul, k '=k(z l ) and A(zl)~A*; since A(zl )~k , A(zl) ~ (AvC~k)*. By the corollary to proposition 1, ~ , and the definition of S, T1, and T2,0~A* for all v, v finite, v r since O~k*, 0 s (A~ c~ k)* for these v. Also by the corollary to proposition 1, ~ , 0 E N k - - / k U (k'~) for all w[ v, v ~ S.

In case v r U1, let D(g) be a basic open subset of Y contained in U 2 such that v e D(g). The inclusion homomorphisms A -~ Ag -~ A~ r~ k -~ Ao induce maps Spec Ao-~ Spec A~ c~ k -~ Spec Ag-~ Spec A. Let Z 2 denote the Ao c~ k-scheme (Proj(~) x r Spec Ag) Xspec A~ x Spec (A~ c~ k).

Z 2 x SpectA~k) Spec A~ is Av-is0morphic to Proj(~)x ~ Spec Av. As in the proof of proposition 9 one sees that Z 2 XSpec{A~r~k)Spec A v and Proj(A~ z2, O) ~ arc isomorphic as A~-schemcs. So it suffices to show that k ' = k(z2) ,

A(z2) ~ (A u n k)* and that 0 ~ (A~ c~ * ' ]v. k) , 0 ~ Nk=/k ~ (kw) for all w By proposition 3, ~ , k '= k(z2) and A(z2)EA*; since A(z2)ek , A(z2)~ ~(Avnk)*. Since vr by the corollary to proposition 1, ~ , 0 e A*; since 0 ~ k*, 0 s (Ao r~ k)*. Also by the same corollary 0 ~ Nk'/k~ (k'~) fo r all w[v.


~ . The zeta function of proj (,~).

Let V be a Severi,Brauer k-scheme of dimension n - 1, n an odd prime, defined by 7 e Br(k). Let X = Proj (.9') be the Y-scheme constructed in w As X is of finite type over Y (proposition 4, w and Y is of finite type over SpecZ, X is of finite type over Spec 2z (EGA I, 6.3.4). For each x ~ X, let o~/m.~ be the residue field of x and let N(X) = card (ox/m~)i Let ,Y denote the set of closed points of X. Then Jf = {x ~ X I N(x) is finite}.

1 (x is defined by (x(S)= l--I ([13]). This product converges

1 - N ( x ) - s

absolutely for Re(s)> n - 1. Let f :X-- , Y be the projection. For each y ~ Y, le t oJ_my be the residue field of y, N(y) = card (oy/my), and X y = f - l ( y ) . Let Y denote the set of closed points of Y. Then (x(s )= = H (x, (s). Each (x, = ((N(Y)-~), where ~ denotes the zeta function of

yeY

Xy regarded as a scheme over oy/m r. Hence ~x contains factors corresponding to each" of the finite places of k.

Recall that for v e S, v finite, X x r Spec A v, is a canonical model for V(Tv) (theorem 1, w and that the zeta function of the closed fiber of a canonical model is given by

r = [(1 -N(v)("- 1-~)")(1 - N ( 1 ) ) ( n - 2 - s ) n ) . . . ( 1 - N(/)) (1 -~)")(1 - N(v)-~)] -1

(corollary of proposition 3, w For all other finite v, X x r Spec Av is either a completed or a degenerate model for a trivial Severi-Brauer kv- scheme and the zeta function of the closed fiber of X x r Spec A ~ is given by

Cx~(S) = [ ( 1 - N(v)n-l-sX1 - N(v) n-z-s)...(1 - N(v) 1 -'X1 - N(v)-S)] -1

(corollaries of propositions 2, 4, and 5, w It is generally agreed that the full zeta function of X should also

contain factors corresponding to each of the infinite places of k, but is not clear just how these factors should be defined. There are two possi- bilities. For an infi'nite place v one could use the factor corresponding to v which occurs in the zeta function Zo(~l of the central division algebra D(7) defined by 7. The other possibility is to use a definition which has been proposed by J. P. Serre (El 2]) and is based on the complex cohomology of oX = ( X xr Spec k) X sp,~k Spec ko. Pursuing either alternative leads to the same definition for the factor corresponding to v, as we now show.

For each infinite v such that ko is isomorphic to R, D(7)| k~ is a central simple R-algebra of dimension n 2 over R. Since n is odd, D(7) | k k~ is kv-isomorphic to Mn(R). Hence the factor in Zo~} corresponding

n--1

to v is I~ G l ( s - i), where Gl(s)=re -'~/2 F(s/2)([15], prop. I, p. 203). For i=0

48 Stuart Turner

each v ~ f~ such that kv is isomorphic to C the factor in Zo(~ corresponding n--I

to v is r [ G 2 ( s - i), where G2(s ) = (2r01-~ F(s). i=0

For each infinite v, ~X can be regarded as an R-scheme in case kv is isomorphic to I~, and as a C-scheme in case k~ is isomorphic to C and an isomorphism between ko and C has been chosen. In either case, let ~X(C) denote the C-valued points of ~f. ~X(C) is a singular C-variety birationally isomorphic to pn-I(c) . Since oX(C) is not a complex analytic manifold, the proceedure proposed by Serre for determining the factor corresponding to v in the zeta function of X is not applicabl e. However, this procedure can be applied to pn-1(C), for which it gives a plausible result.

For each i,'0 < i < 2(n-1) ,Hi(pn-I (C) ,C) is a C-vector space of dimension one or zero, according as i is even or odd. In case i is even, the Hodge decomposition of Hi(pn-I(C), C) is simply ~i/2. i/2. If k v is isomorphic fo C, the factor corresponding to v for the i th cohomology group is (2n)-1 Gz(s _ i/2) or 1, according as i is even or odd; the complete factor corresponding to v is Gc(s ) = [-I (2r0-i G2(s- i/2). This is

0<i_<2(n- 1) i, even

the same factor as the one corresponding to v in ZD(y), except for the innocuous (27r) -n. If k~ is isomorphic to ~ G = Gal (C/R) acts on 0 ~ - 1(C) by conjugation, al~ anti-analytic automorphism. So G acts on each HI(pn-I(C), C). Let g be the non-trivial element of G and x ~ H~(/~-I(C), C), x :~ 0, then g(x)= ( - 1) i/2 x. Thus, the factor corresponding to v for

t h e i 'h cohomology group is Gx(s - i/2) or 1, according as i is even o r odd. The complete factor corresponding to v is G~s) = I-[ G l ( s - i/2),

O<i<2(n - - l ) i, e v e n

the same factor as the one corresponding to v in Zo(~).

Definition. Let r t be the number of real places of k and r 2 be the number of complex places of k. The function Zx(s ) = Ga(s)"Gc(s) "2 (x(S), defined in the half-plane Re(s) > n and homomorphic there, will be called the full zeta function of X.

Proposition 1. Let s ~ C, Re(s )> n: Then

Z~(s) = = Zo(~,(s)" [-I ['(1 - N(v)Cn-I-~n)(1 - N(v) ("-2-~") ... (i - N(v) '1- ~"'l - l .

v6S v, f i n i t e

Proof. Since n is prime, for each v ~ S, o finite, D(?) | k~, is a central division algebra over kv and the factor corresponding to o in Zo~r~ is (1 - N(v)-~) - 1


([15], proposition 7, p. 197). For all other finite v the factor corresponding to v in Zorn,)is [ ( 1 - N(v)" - l - s ) (1 - 'N(v)n-2-s ) . . . (1 - N(v)-~)] -1. So Z x and Zo(v) are formed from the same local factors for all infinite ~ and for all finite v,v r S. The conclusion follows by comparing the factors in Z x corresponding to each v a S, v finite, with their counterparts in Zo~;.~.

Theorem 1. Z x can be continued analytically as a meromorphic jhnction in the s-plane, holomorphic except for simples poles at s = 0 and s = n. Z x satifies the functional equation

Z s) = " l 0 1 - s),

where N6 is the, norm of the different of D(y) over k and D is the discriminant ofk. Proof. Zotr~ is a meromorphic function in the s-plane, holomorphic except for simple poles at s = 0 and s = n ([14], theorem 2, II). The finite product which appears in the statement of proposition 1 defines a non-vanishing holomorphic function in the s-plane, so the first two assertions follow from proposition 1. Zotr~ satisfies the functional equation

Z~(s ) = (N~) 1/2-'/"1 D ] n~n/2-~ Zo~,(n - s)

([14], loc. tit.). So by proposition l

Zx(s) = ~ (N(vr-"/2) "~"-1)" (N6)'/2-s/nt D I n'n/2-~'Zx(n - s).

v..finite

For v ~ S, v finite, N6o, the norm of the different of D(y) (~k kt. over k r, is given by N6 v = N(v)ntn-l~([14]), proof of proposition 2, II). As N6 = = ~ N6v, the functional equation has the form asserted.

v, f inite

50 Stuart Turner

References

1,1] A. Albert, Structure of Algebras, Amer. Math. Soc. Colloquium Publications, XXIV, 1939.

1,2] S. Amitsur, Generic splitting fields of central simple algebras, Ann. of Math. 62 (1955), 8-43.

1"3] E. Artin and J. Tate, Class Field Theory, Benjamin, 1967. [4] Z. Borevich and I. Shafarevich, Number Theory, Academic Press, 1966. [5] J. W. S. Cassels and A. Fr6hlich, ~ds., Algebraic Number Theory, Thompson Book

Company, 1967. 1,6] F. Chatelet, Variations sur un thdme de H. Poincard, Ann. ENS, 59 (1944), 249-300. [7] M. Deuring, Algebren, Springer-Verlag, 1968. 1"8] H. Hasse, Zahlentheorie, Akademie-Verlag, 1969. 1"9] S. Lang, Algebraic Number Theory, Addison-Wesley, 1970.

[10] P. Roquette, On the Galois cohomology of the prqjective linear #roup .... Math. Ann. 150 (1963), 411-439.

C11] J.-P. Serre, Corps locaux, Hermann, 1968. 1"12] J.-P. Serre, Facteurs locaux des fonctions z~ta des varidtds algebriques (d6finitions

et conjecttires), S6minaire Delange-Pisot-Poitou, 1969/70, exp. 19. [13] J.-P. Serre, Zeta and L-functions, in Arithmetical Algebraic Geometry, Harper and

Row, 1965, 82-92. 1,14] S. Turner, Zeta-functions of central simple algebras over globat fields, An. Acad. Brasil.

Ci. 48 (2), (1976), 171-186. [15] A. Weil, Basic Number Theory, Springer-Verlag, 1967. [16] O. Zariski and P. Samuel, Commutative Algebra, Vol. I, D. Van Nostrand, 1958.

A.C.N. Bourbaki, Elements de Mathematique, Alodbre Commutative, Hermann, 1961. EGA. A. Grothendieck and J. Dieudonn6, Elements de G#om&rie Alg~brique, I. Springer- Verlag, 1971; II, IV, Publications Math6matiques, IHES, 1961-65.

School of Mathematics, The Institute for Advanced Study, Princeton, New Jersey, 08540. Departamento de Matem~tica, Pontificia Uni- versidade Cat61ica do Rio de Jaaeiro, Run Marqu6s de Sgo Vicente 209/263, 22453 Rio de Janeiro, Brazil.

the zeta function of a birational severi-brauer scheme

Documents