Euler-Poincaré Characteristics for Motivic Cohomology and Arithmetic Groups

Euler-Poincaré Characteristics for Motivic Cohomology and Arithmetic GroupsAuthor(s): Stuart TurnerSource: American Journal of Mathematics, Vol. 114, No. 3 (Jun., 1992), pp. 657-665Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2374774 .

Accessed: 16/12/2014 23:28

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 192.231.202.205 on Tue, 16 Dec 2014 23:28:39 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/2374774?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


EULER-POINCARE CHARACTERISTICS FOR MOTIVIC COHOMOLOGY AND ARITHMETIC GROUPS

By STUART TURNER

Euler-Poincare characteristics have a way of cropping up when one studies the values of zeta functions at integers. On the one hand, they arise in arithmetic versions of the Gauss-Bonnet theorem [On], [H], [S], [T2], and, on the other, in applications of etale cohomology and of K- theory to varieties over finite fields [L1-4], [BN], [Sch], [M2-3]. Here we investigate some consequences of the Lichtenbaum-Milne theory and of Serre's theory of cohomology of S-arithmetic groups as they apply to elliptic surfaces over finite fields.

Some of Serre's definitions are recalled briefly at the beginning of Section 1. This introduction is followed by a calculation of the Euler- Poincare characteristic of a v-arithmetic group of units in a maximal order of a quaternion algebra over a global field of positive characteristic. In Section 2 we show how to associate such a "quaternion group" to each irreducible fiber of an elliptic surface defined over a finite field and carrying a level two structure. The question then is how the Euler characteristics of these groups relate to the arithmetic of the surface. Their product diverges, but if each Euler characteristic is appropriately weighted, the product converges to the motivic Euler characteristic of the constant sheaf Z in the etale topology. (The statement of the theorem is not quite this clean as it is necessarily smudged by the presence of some global constants and by factors coming from the singular fibers. Two remarks following the theorem indicate how it might be polished.)

The paper closes with an appendix that studies in detail the asso- ciation of an algebraic family of quaternion algebras to the smooth fibers of an elliptic surface with level two structure. This construction may be of independent interest for the study of the class field theory of the surface.

Manuscript received 18 May 1990. Partially supported by CNPq. American Journal of Mathematics 114 (1992), 657-665.

657



658 STUART TURNER

1. A discrete group F is said to be of type (FL) if Z, regarded as a Z(F) module, has a finite free resolution 0 - > ... * -> 0 [S] Section 1.2. The Euler-Poincare characteristic x(F) of F is then defined by x(F) = 1( - 1)1 rkrLi and it is easy to see that x(F) is independent of the finite free resolution chosen. Serre defines a group F to be of type (VFL) if it has a subgroup of finite index of type (FL), op. cit. Section 1.8. For such a group let F' be a subgroup of finite index of type (FL) and define, following Wall [Wa], x(F) = [F F']`-X(). Again, it is easy to see that x(F) does not depend on the choice of F'. There is a stronger notion of a group of "type (WFL)" which requires that F be "virtually torsion free" and that all torsion-free subgroups of finite index in F are of type (FL).

Let G be a unimodular locally compact group and p. be an invariant signed measure on G. For a discrete subgroup F denote also by p. the induced measure on GIF. Serre op. cit. Section 3.1, calls p. an Euler- Poincare measure if for every torsion-free discrete subgroup F with compact quotient GIF we have i) F is of type (FL) and ii) X(F) = fG/r

[L. This says, in effect, that the Euler-Poincare characteristic of F, which is defined by a finite free resolution of F, can be expressed in measure theoretic terms. An Euler-Poincare measure, if one exists, is unique.

In his monograph Serre deals primarily with Lie groups, both real and ultrametric, and with S-arithmetic subgroups of connected reductive groups over global fields. The latter case is central for this paper, but as the Lie group case provides both motivation and intuition it seems best to mention it briefly before getting down to work. The fundamental example occurs when G is a unimodular real Lie group with finitely many connected components and F is a torsion-free discrete subgroup with compact GIF. For a maximal compact subgroup K of G let GIK be the homogeneous space endowed with a G-invariant Riemannian metric. Denote by w the unique invariant measure on G which induces the measure defined by the curvature form on GIK. A simple application of the Gauss-Bonnet-Chern theorem to F\GIK shows that w is an Euler- Poincare measure on G, op. cit. Section 3.2. We now turn to the arithmetic case.

Let K be a global field of genus g with field of constants Fe, v be a place of K, and R(v) be the subring of K consisting of elements which are integral at all places w of K, w =# v. Let H be a quaternion algebra over K, not isomorphic to M2(K) and unramified at v, H' be the elements of H with reduced norm equal to one, e be a maximal R(v)-order in H,



EULER-POINCAREI CHARACTERISTICS 659

and Cl = Hfl nO. We now compute X(VY) using the Euler-Poincare and Tamagawa measures on SL2(K,).

Let G be the semisimple K-algebraic group associated to H'. G has K-rank zero and Cl may be regarded as a v-arithmetic subgroup of G(K) = H'. By [S] Section 2.4, Theorem 4 Cl is a group of type (WFL). Let GA be the adelization of G and embed G(K) diagonally in G. For a place w of K let Rw and Few denote, respectively, the ring of integers and the residue field of Kw. Set Ow = e ?R(v) Rw and denote by O' the elements of Ow with reduced norm one. Let C = Hlwsv O and U SL2(Kv) x C. U is an open subgroup of GA, GA = G(K) U and G(K) n u = 6', so UIi' is isomorphic to GAIG(K). G(K) is a discrete subgroup of GA with compact quotient [W1] Chapter III, Lemma 3.1.1, [W2] Chapter IV Section 3, Theorem 4, and hence Cl, embedded diagonally in U, is a discrete cocompact subgroup of U.

Letp denote the projection of C' on the factor SL2(Kv) of U. Clearly, p(C') is isomorphic to C' and by [V] IV Section 1, Lemme 1.2, p(Y') is a discrete subgroup of SL2(Kv) with SL2(Kv)lp(Q') compact.

PROPOSITION. X(61') = W3 3_K(2)(1 - iv) Hw D 112(l - i '), where Dw = f2 is the discriminant of Hw and the product extends over all places of K where H is ramified.

Proof. The Euler-Poincare measure l.v on SL2(Kv) is the invariant measure such that 1iv(SL2(Rv)) = 1 - tv [S] Section 3.3. By op. cit. Section 3.1 Remarque 1, x(p(C')) = [v(SL2(Kv)Ip(Y')). In order to compute this last quantity first observe that UKY'C and SL2(Kv)lp(C') are isomorphic. Since UIi' is isomorphic to GAIG(K) we transport the Tamagawa measure on GAIG(K) to UIi' and then compare '(UAC'C) with 1iv(SL2(Kv)Ip(Y')).

Recall the definition of T. For a place w of K where H is unramified let w : Hw -> M2(Kw) be a Kw-isomorphism. The local Tamagawa measure ww on H' = G(Kw) is the Haar measure which gives the compact subgroup fw-'(SL2(Rw)) volume 1 - i?-2. Since all Kw-automorphisms of M2(K) are inner, this definition does not depend on the choice of fpw. For a place w where Hw is ramifiod, Hw is a division algebra and the local Tamagawa measure on HW is the Haar measure which gives the group of units in the unique maximal order volume D- 1/2(1

iW-2)(1 - c'-l)-', where Dw = 2 is the discriminant of Hw [T1] II, Proposition 8, [V] III, Section 2. The global Tamagawa measure T on GA is 0g3 times the product measure Hrw w and 7(GAIG(K)), the Tam-



660 STUART TURNER

agawa number of G, is one [V] loc. cit., [W1] Chapter III Sections 5, 6.

By the isomorphisms above '(SL2(Kv)Ip(Q')) = '(C)-' = (Hov 0w)-1 = f3-3g H1+O (1 - p-2)-l 1Dl/2(1 _ e-2)-1(1 _i?-') where the first product extends over all w -such that Hw is unramified and the second over all w such that Hw is ramified. Thus, 'r(SL2(Kv)Ip(tY')) =

3 - 3%-K(2)(1 _

f-2) flw D1'2(1 - te2). Since wv = (1 _ -2)(1 -

tv)-'1lv, 1iv(SL2(Kv)Ip(0')) = f3-3 _3(2)(1 - ev) nl D1'2(1 - e-1).

2. Let C be a complete, smooth, geometrically irreducible curve over Fq, q odd, L be the function field of C, (E, o) be an elliptic curve over L, and assume that the 2-torsion points of E are L-rational. Let Z denote the complete Neron model of E, p: -> C be the Fq-rational projection, and o,: C --> be the section corresponding to o.

For each place v of L let qv = qdeg . Except for a finite number of v, the fiber p -(v) is smooth and may be regarded as an elliptic curve over Fqv. Other fibers may be singular or reducible. Whether the fiber is singular or not, since the characteristic is different from two, the IFq- rational sections vi, i = 0, 1, 2, 3 corresponding to the 2-torsion points on the generic fiber intersect the fiber p (v) in four distinct points [N] II Section 9, Lemme 2.

Let c,p-1(v)),c0 denote the local ring of the component of p-'(v) containing uo-v and K(Zv, o,) be its residue field, then K(Zv, o,) is isomorphic to the function field of this component of the fiber and has constant field Fqv. If the fiber is singular, K(Zv, o,) is a field of genus zero, and, since the field of constants is finite, has a place of degree one; it is thus a rational function field.

For each place v of L such that p-'(v) is irreducible let Av denote the quaternion algebra over K(Zv, ,,) ramified at vi(v) n p- (v), i = 1, 2 and unramified at all other places of K(v, o,). By our assumptions these two places are of degree one. Let S(v) be the subring of K(v, ao,) consisting of functions whose only poles are at u0(v) n p-i(v), MV be a maximal S(v)-order in Av, MV be the set of elements of MV with reduced norm equal to one, and Xv(Ml) be the Euler-Poincare characteristic of Ml. We wish to study a weighted product of the Xv(Ml).

Let 4(Z, s) and t(Zv, s) denote the zeta functions of Z and of Cv, respectively. For Re s > 2, flv 4(Wv, s) converges to 4(W, s). Let tK(Evxro)(S)

be the zeta-function of K(cv, o,) and write ~K(%,,o) = Pv(q-s)(1 -

q-s)-l(1 - ql-s)-l. If ev is smooth, Wkv S) = UK(,o)(s), so (e s)



EULER-POINCARE CHARACTERISTICS 661

Hrv K(%V,u)(S) Hw Kw(s), where the first product is taken over all fibers of p, Kw(s) = t(Zw, S)UK('6,cJO)(S)-1, and the second product extends over all singular fibers. Let t(C, s) denote the zeta function of C, then t(w, s)t(C, s - 1)-1 = t(C, s) flv Pv(q-s) flw Kw(s), where the product converges absolutely for Re s > 3/2. Recall that t(W, s) and t(C, s -

1) both have simple poles at s = 2.

PROPOSITION. lim,2 ( S)t(C, S - 1)-1 = Hv qv3(1 -

q7-1)-2jX,(M1)j. Hw q-3KKw(2). H, 4(%x, 2) (1 - q-1), where the first product extends over all irreducible fibers of p, the second over singular irreducible fibers, and the third over all reducible fibers.

Proof. Since K(Zv, a,) has constant field Fqv and the two places where Av is ramified have degree one, by the proposition in Section 1,

XV(MV)= q5v-3gvU( 6v,u)(2)(qv - 1)(1 - q2-1)2, where gv denotes the

genus of K(v, a,,). Hence Hlv q;-3(1 - q;-1)-21XV(M) = t(C, 2) Hx (1 - qx2) Hlv q3v- vPv(q -2). For all v such that p'(v) is smooth, gv = 1,

so the proposition follows from the product expansion for t(e, s)t(C, s

Remark. Since t(W, s) satisfies the functional equation t(e, 2 - s) = + q(l -S)C2t4(, s), where c2 is the second Chern class of the tangent bundle on Z, lims52 t(W s)(C, s - 1)-l = + qC2 lim5-2 t(, 2 - SWC, s - i)-1. We now express this limit in terms of motivic cohomology.

Let H'(W, 7) denote the etale cohomology group of the constant sheaf 7. Then H0(W, 7) -Z and H'(W, Z) is torsion for i > 0, [M2] Corollary 7.2. For i - 3, Hi(W, Z) is finite and is isomorphic to Hi(W, 7Z) = Ie H(W, Z/e) where H'(W, Z/e) = lime Hi(W, Z/eln) and the product is taken over all primes 4 including 4 = p, [M2] Corollary 7.6(a). Hence Hi(W, 7) = 0 for i > 5, [Ml] VI, Corollary 1.4 and Remark 1.5(b). H'(W, 7) is also finite, [M2] Lemma 9.1, and H2(, ) H2(, )cotor ED QI/ where H2( , Z)cotor iS the cotorsion quotient group of H2(Z, 7), [M2] Remark 7.7(a). Let

c [(RZ 'SHI[1K( 'S0H][(K 0'( H] _ O)0HX [(Z C 'SH[OAP010(z C )ZH][P?l(Z 'SOoH

where [G] denotes the order of a group G. Theorem 0.4(a) of [M2] asserts that lim,0 (1 -

q-S)t(W, s) = ?+X(Z, 7). (Note the obvious

misprint in the statement on p. 301.)



662 STUART TURNER

LEMMA. Let h = [Jac C(Fq)], then lim,2 4(W, s)4(C, S - 1)-i =

+qC2X(Z, 7)h-lqg-l(q - 1).

Proof. By the remark it suffices to show that lim,2 (, 2 -

s)t(C, s - 1)-1 = x(W, 7)h-lqg-l(q - 1), but this follows immediately from Milne's theorem and [W2] Chapter VII Section 6, Theorem 4.

THEOREM. X(% 7/) = ql c2h(q - 1)-i H q-3(1

v- 1) - 21 Xv(M v) |. - 3K(2). H, 4(%xi 2)(1 - q- 1), where thefirst product

extends over all irreducible fibers of p, the second over all singular irreducible fibers, and the third over all reducible fibers.

Proof. Since both sides of the formula are positive, the theorem follows from the proposition and the lemma.

Remarks.

1) By Milne's theorem, the motivic Euler-Poincare characteristic of C has value X(C, 7) = hlq - 1.

2) The theorem gives an expression for x(W, 7) as a product of factors associated to the fibers of p. Recall that q -C2 can also be decom- posed in the same way. Let mv denote the number of components of p-'(v), Ev = 0, 1, or 2 depending on whether the fiber is smooth, of multiplicative type, or of additive type, and bv be the "measure of wild ramification" (5v = 0, unless the fiber is additive type and the residue field characteristic is two or three), then c2 = Ev mv + Ev - 1 + 5v.

For details see [0] IV.

Appendix. The set of quaternion algebras Av defined in Section 2 by the level two structure (0o1, 02) on Z may be regarded as an algebraic family parametrized by the points v of C such that p- 1(v) is irreducible. Over the subset of v's such that p -1(v) is smooth these algebras can be written as a family of cyclic algebras split by an algebraic family of unramified extensions of the fields K(v, a,,) determined by the section u3. In the first two sections of this appendix we give the construction on one fiber and then treat the global case in the third.

1. Unramified quadratic extensions associated to points of order two. Let K be a field of genus one with field of constants F1, 2 { , v0 be a place of degree one, and E be an elliptic curve over Fe with distinguished point v0. Multiplication by two on E defines an isogeny




of degree four; denote by K'IK the extension corresponding to this isogeny. Let KA* and KA* denote, respectively, the idele groups of K and K', KX be the ideles of K of module one, and U = K*NK,,K(KA*) be the subgroup of KA* associated to K' by class field theory. Let R,. denote the valuation ring of K,0 and ar,o be a prime element of R,O. The sequence 1 -> KA -> KA {-i} -> 1 defines an isomorphism KAVKA n u KAIU and splits by the map t -- w-r', so KA* = KA x {-Trr0}. Let uo =

1, U1, U2, U3 be the representatives of the cosets of KA n U in KA, so 7rvoui, i = 0, 1, 2, 3 are representatives of the cosets of U in KA. Let N1 = {rvou1} x U; the Ni, i = 1, 2, 3, are the subgroups of KA* corresponding to the three quadratic extensions Ki, i = 1, 2, 3 of K contained in K'.

LEMMA. Let v be a place of K of degree one, then v splits in precisely one of the fields Ki, i = 1, 2, 3.

Proof. The places of Ki above v are in one-to-one correspondence with the cosets of K* Ni in KA* [W2] Chapter XIII Section 10, Proposition 14, so v splits in Ki if and only if [K*: K* Nj] = 2. However, [KA: Nj] = 2 so v splits completely in Ki if and only if K* C Ni. Since KilK is unramified, R*, the units in the valuation ring of Kv, is contained in Ni, i = 1, 2, 3 loc. cit. Corollary 2 to Proposition 14. Let Trv be a prime element in Rv, then Trv E Ni if and only if wvw`r17r E uiU. There is a unique i for which this is the case. U

Let vi, i = 1, 2, 3 be the three points on E of order two and assume that they are of degree one. Reindex the fields Ki, i = 1, 2, 3 so that Ki is, by definition, the field in which the place vi of K splits. Thus, 7r.7rV' E ujU, and the elements Trv,-rTr', i = 0, 1, 2, 3 form a set of representatives of the cosets of KA n U in KA and also of the cosets of U in KA. We summarize some of the results in the following

PROPOSITION. There is a canonical one-to-one correspondence be- tween the places vi, i = 1, 2, 3 and the quadratic extensions Ki of K contained in K'. A place vi corresponds to a field Ki if and only if Ki is the class field associated to the subgroup {-rv,ar 'I} x U of KA*. The place vi splits in Ki and the two places vj, j =# i, are inert in Ki.

2. The cyclic algebra determined by a -level two structure. The places v1 and v2 define a level two structure on E. We use this structure to construct a quaternion algebra over K that is split by K3 and ramified only at v1 and v2. Let ED denote the group law on E, then v1 flV2 = V3,



664 STUART TURNER

so the divisor v1 - vo + v2 - v0 is linearly equivalent to v3 - vo. Take f E K such that div(f) = V1 + V2 - v3 - vo. Let X be the nontrivial character on Gal (K31K) and H be the cyclic algebra [K31K; X, f], [W2] Chapter IX Section 4. Since v1 and v2 are inert in K3 and ordv,f = ordv2f = 1, f is not a norm from the places of K3 lying over v1 and v2, so H is ramified at v1 and v2, i.e., H ? Kv, i = 1, 2, are division algebras. On the other hand, v3 splits in K3 and ordvf = -1 so f is a norm from the places of K3 lying over v3. Since v0 splits completely in K' and therefore splits K3, f is also a norm from these places of K3. K31K is unramified and ordvf = 0 at all other places v of K, so f is a norm in all these Kv*. H is therefore a quaternion algebra unramified at all places of K except v1 and v2. Observe that the element f is determined up to multiplication by a nonzero element of the constant field and that this is the only point in the construction where a choice has been made.

3. Families of cyclic algebras over smooth elliptic fibrations. Now we return to the situation treated in Section 2: Z is a relatively minimal elliptic surface over C with zero-section o,, and three Fe-rational sections vi, i = 1, 2, 3 of order two. Here we consider only the smooth fibers of the projection p. The pair (U1(v), U2(v)) determines a level two structure on p-1(v) and, by Section 1 of the Appendix, an unramified quadratic extension K(Zv, U0)31K(Zv, o,). Assume there is an Fe-rational section : C --> which does not intersect the sections vi, i = 0, 1, 2, 3, for example, an Fe-rational section of order four. For each v there is thus a unique fv E K(Zv, o7,) such that div(fv) = u(v1) + (J(V2) - u(v3)

- U(v0) and fv(T(v)) = 1. The quaternion algebras Av defined in Section 2 have a canonical representation as cyclic algebras [K(Zv, ffo)31K(Wv,

U'), Xv, fv], where Xv is the nontrivial character on Gal (K(Zv, u0)31K(Wv, o,)). Hence, the Av are not merely determined "up to isomorphism," but form a family "rigidified" by the level two structure on Z and the section T. Since the fv cannot be lifted to c0,p-i(v),,0 ), the Av do not appear to arise from an Azumaya algebra on W. It seems more likely that they are related to the higher dimensional class field theory of Bloch, Kato, Parshin and Saito.

PONTIFlCIA UNIVERSIDADE CAT6LICA DO RIO DE JANEIRO, BRASIL




REFERENCES

[BN] P. Bayer and J. Neukirch, On the values of zeta functions and ?-adic Euler characteristics, Inventiones Math., 50 (1978), 35-64.

[H] G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Scient. Ec. Norm. Sup., 4a. serie, t.4 (1971), 409-455.

[Ll] S. Lichtenbaum, On the values of zeta and L-functions: I, Ann. of Math., 96 (1972), 338- 360.

[L2] , Values of zeta-functions, etale cohomology, and algebraic K-theory, in "Classical" Algebraic K-Theory and Connections with Arithmetic, Lecture Notes in Mathe- matics, 342, Springer-Verlag, Berlin, Heidelberg, New York (1973), 489-501.

[L3] , Zeta-functions of varieties over finite fields at s = 1, in Arithmetic and Geometry, Papers Dedicated to I. R. Shafarevich on the Occasion of his Sixtieth Birthday, vol. I Arithmetic, eds. Michael Artin and John Tate, Progress in Mathematics 35, Birkhauser, Boston, Basel, Stuttgart (1983), 173-194.

[L4] , Values of zeta-functions at nonnegative integers, in Number Theory Noordwijk- erhout 1983, Lecture Notes in Mathematics, 1068, Springer-Verlag, Berlin, Hei- delberg, New York, Tokyo (1984), 127-138.

[Ml] J. S. Milne, Etale Cohomology, Princeton University Press, Princeton (1980). [M2] , Values of zeta functions of varieties over finite fields, Amer. J. Math., 108 (1986),

297-360. [M3] , Motivic cohomology and values of zeta functions, Compositio Math., 68 (1988),

59-102. [N] A. Neron, Modeles minimaux des varietes abeliennes sur les corps locaux et globaux, Inst.

Hautes Etudes Sci. Publ. Math., 21 (1964), 361-484. [0] A. Ogg, Elliptic curves and wild ramification, Amer. J. Math., 89 (1967), 1-21. [On] T. Ono, On algebraic groups and discontinuous subgroups, Nagoya Math. J., 27 (1966),

279-322. [Sch] P. Schneider, On the values of the zeta function of a variety over a finite field, Compositio

Math., 46 (1982), 133-143. [S] J. P. Serre, Cohomologie des groupes discretes, in Oeuvres Collected Papers, vol. II, Sprin-

ger-Verlag, Berlin, Heidelberg, New York, Tokyo (1986), 593-685. [Tl] S. Turner, Zeta-functions of central simple algebras over global fields, An. Acad. Bras.

Cien., 48 (1976), 171-186. [T2] , A Gauss-Bonnet theorem for motivic cohomology, Inventiones Math., 101 (1990),

57-62. [V] M. F. Vigneras, Arithmetique des Algebres de Quaternions, Lecture Notes in Mathematics,

800, Springer-Verlag, Berlin, Heidelberg, New York (1980). [W] C. T. C. Wall, Rational Euler Characteristics, Proc. Cambridge Phil. Soc. (Mathematical

and Physical Sciences), 57 (1961), 182-184. [Wl] A. Weil, Adeles and Algebraic Groups, notes by M. Demazure and T. Ono, Progress in

Mathematics, 23, Birkhauser, Boston (1982). [W2] , Basic Number Theory, Die Grundlehren der Mathematischen Wissenschaften

144, Springer-Verlag Berlin, Heidelberg, New York (1967).



Euler-Poincaré Characteristics for Motivic Cohomology and Arithmetic Groups

Documents

Transcript of Euler-Poincaré Characteristics for Motivic Cohomology and Arithmetic Groups