Dedekind zeta motives for totally real fields

arXiv:0804.1654v1 [math.AG] 10 Apr 2008

FRANCIS C.S. BROWN

Dedekind zeta motives for totally real fields. Abstract. Let k be a totally real number field. For every odd n ≥ 3, we construct a Dedekind zeta motive in the category MT(k) of mixed Tate motives over k. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of π n[k:Q] ζk∗ (1 − n), where ζk∗ (1 − n) denotes the special value of the Dedekind zeta function of k. We deduce that the group Ext1MT(k) (Q(0), Q(n)) is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for k that have been conjectured to calculate the groups Ext1MT(k) (Q(0), Q(n)). In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over k.

Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction Spaces of constant curvature Volumes of product-hyperbolic manifolds and L-functions Triangulation of product-hyperbolic manifolds The motive of a product-hyperbolic manifold Applications Example: A Coxeter motive for L(χ, 3) Bibliography

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1 9 11 21 34 53 60 62

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FRANCIS C.S. BROWN

1. Introduction 1.1. Motivation and history. This paper was motivated by a desire to reconcile two important and complementary results due to Zagier and Goncharov. Firstly, in 1986, Zagier obtained a formula for the value ζK (2), where K is any number field, as a linear combination of products of values of the dilogarithm function at algebraic points [65], giving a spectacular generalization of Dirichlet’s class number formula. The idea is to compute the volume of an arithmetic hyperbolic manifold in two different ways. Suppose that K is quadratic imaginary, and let Γ be a torsion-free subgroup of finite index of the Bianchi group PSL2 (OK ). Then Γ acts on hyperbolic 3-space H3 , and a classical theorem due to Humbert states that |d|3/2 n ζK (2) , 4π 2 where d is the discriminant of K, and n is the index of Γ in PSL2 (OK ). On the other hand, a hyperbolic manifold can be triangulated using geodesic simplices, whose volume, by a calculation originally due to Milnor, can be expressed in terms of the Bloch-Wigner dilogarithm function X zn . D(z) = Im (Li2 (z) + log |z| log(1 − z)) , where Li2 (z) = n2 vol(H3 /Γ) =

n≥1

3

The volume of H /Γ can therefore also be expressed as a sum of values of dilogarithms. Equating the two volume calculations yields Zagier’s formula. This in turn motivated his conjectures relating polylogarithms to special values of zeta and L-functions [28]. In the original paper [65], he also obtained a formula for ζK (2) for any number field by considering groups acting on products of hyperbolic 3-space. However, one problem is that the dilogarithms are evaluated over some finite extension of K, rather than K itself. Another problem is that it is far from clear how one could prove that the sums of products of dilogarithms are in fact a determinant, as should be the case. The results of Zagier were subsequently reinterpreted in terms of algebraic Ktheory by a large number of authors, which we will not attempt to list here. Essentially, the gluing equations between simplices imply that an (ideal) triangulation of a hyperbolic manifold defines an element in the K-theory group K3 (Q) ⊗ Q, via the so-called Bloch group. The dilogarithm can be interpreted as the regulator map. Some of these ideas are surveyed in [8, 28, 44]. The connection with the theory of mixed Tate motives emerged in the papers [5, 6]. Secondly, in the paper [29], Goncharov generalized these constructions for any discrete torsion-free group Γ of finite covolume acting on hyperbolic n-space Hn , for n = 2m − 1 odd. For any such manifold M = Hn /Γ, he constructed an element ξM ∈ K2m−1 (Q) ⊗ Q , such that the volume of M is given by the regulator map on ξM . In fact, he gave two such constructions, but it is the motivic version which is of interest here. The main idea is that a finite geodesic simplex in hyperbolic space defines a mixed Tate motive. By triangulating the manifold M , and combining the motives of each simplex, one obtains a total motive for M . If M is non-compact, he uses the Sah isomorphism to replace the original triangulation with an equivalent, finite one. He then proves that the motive of M is an extension of Q(0) by Q(n), and identifies

DEDEKIND ZETA MOTIVES FOR TOTALLY REAL FIELDS.

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the group of such extensions with K2n−1 (Q) ⊗ Q. The proof of this fact is analytic: it uses the fact that angles around faces in a triangulation sum to multiples of 2π, and uses the full faithfullness of the Hodge realization. In this paper, we give a common generalization of both Zagier and Goncharov’s results. Firstly, we give a complete list of arithmetic groups acting on products of hyperbolic spaces Hn1 × . . . × HnN , and compute their covolumes, up to a rational multiple, in terms of zeta and L-functions. This uses well-known Tamagawa number arguments. Next, we show that any complete product-hyperbolic manifold M of finite volume admits a triangulation using products of geodesic simplices, and use this to construct a well-defined mixed Tate motive mot(M ). In the case N = 1 when there is a single hyperbolic component, our construction is different from Goncharov’s in the non-compact case and avoids the use of the Sah isomorphism for hyperbolic scissors-congruence groups. We then prove that mot(M ) ∈ Ext1 (Q(0), Q(n1 )) ⊗ . . . ⊗ Ext1 (Q(0), Q(nN ))

using a purely geometric argument. Thus the construction is completely motivic. Let k be a totally real number field, and n an odd integer ≥ 3. The Dedekind zeta motive is obtained in the case where Γ is a torsion-free subgroup of the restriction of scalars Rk/Q SO(2n − 1, 1), and its period is directly related to ζk (n). In this case, we prove that mot(M ) is in fact a determinant. Using Borel’s calculation of the rank of algebraic K-groups, we deduce that mot(M ) is a multiple of the determinant of the algebraic K-theory of k. This gives generators for K2n−1 (k) ⊗ Q in terms of hyperbolic geometry and proves, in particular, that the zeta value is a determinant of volumes of hyperbolic simplices, which seems to be inaccessible via a purely geometric argument. Before stating our main results in greater detail, we give a brief digression on mixed Tate motives and elementary motivic complexes. 1.2. Motivic complexes. Let k be a number field. There exists an abelian tensor category MT(k) of mixed Tate motives over k, whose simple objects are the Tate motives Q(n), for n ∈ Z, and which is closed under extensions [22]. The structure of this category is determined by its relation to algebraic K-theory: 0 if n ≤ 0 , (1.1) Ext1MT(k) (Q(0), Q(n)) ∼ = K2n−1 (k) ⊗ Q if n ≥ 1 , and the fact that all higher extension groups vanish [22]. By a theorem due to Borel [10], one has K1 (k) = k × , and for n > 1, r1 + r2 , if n is odd ; (1.2) dimQ (K2n−1 (k) ⊗ Q) = n± = r2 , if n is even ,

where r1 is the number of real places of k, and r2 is the number of complex places of k. The Hodge realisation functor on MT(k) gives rise to a regulator map (1.3)

rH : Ext1MT(k) (Q(0), Q(n)) −→ Rn± .

The importance of the regulator is in its (largely conjectural) relation to the special values of Artin L-functions [3, 9, 23, 41]. There have been various attempts to construct categories of mixed Tate motives from the bottom up, out of simple geometric building blocks. In particular, the groups (1.1) should be the cohomology groups of certain motivic complexes, and should therefore be given by explicit generators and relations. This leads to the

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FRANCIS C.S. BROWN

following problem: to construct elements in Ext1MT(k) (Q(0), Q(n)) out of the cohomology of simple algebraic varieties, and calculate the image under (1.3). This problem can be formulated in a precise manner in at least three different ways. 1.2.1. Hyperplanes. The approach of [6, 7] uses the relative cohomology of arrangements of hyperplanes in projective space. Consider a pair of simplices (L, M ) in Pn , defined by a set of 2n + 2 hyperplanes L0 , . . . , Ln , M0 , . . . , Mn ⊂ Pn in general position. If (L, M ) are defined over k, such a pair defines a mixed Tate motive: (1.4) H n Pn \M, L\(L ∩ M ) ∈ MT(k) .

The scissors-congruence group for hyperplanes, which we denote Snh (k), is defined by taking the Q-vector space generated by all (admissible) pairs (L, M ) defined over k, modulo a certain number of relations, of which the most important are: n+1 X i=0

b j , . . . , Ln+1 , M ) = 0 , (−1)i L0 , . . . , L

along with a similar equation for M instead of L, and the relation (gL, gM ) = (L, M ) for all g ∈ PSLn+1 (k). If one considers hyperplanes in general position only, one can show that the graded vector space S•h (k) has a coproduct [66]. Let H 1 (Snh ) denote the subspace of primitive elements of degree n. Then one obtains a map (1.5)

φh : H 1 (Snh (k)) −→ Ext1MT(k) (Q(0), Q(n)) .

A conjecture in [6] (see [65]), in the number field case, states that the map φh is an isomorphism. In particular, its surjectivity would imply that (1.1) is generated by motives of hyperplanes (1.4). 1.2.2. Quadrics and hyperplanes. Another approach to the same problem was proposed by Goncharov in [29]. Let Q denote a smooth quadric in P2n−1 , and let L1 , . . . , L2n denote 2n hyperplanes in general position with respect to Q. If the pair (Q, L) is defined over k (and if the discriminant of Q is in k ×2 ), this defines what we shall call a quadric motive: (1.6) H 2n−1 P2n−1 \Q, L\(Q ∩ L) ∈ MT(k) .

The vector space of Q-linear combinations of pairs (Q, L) defined over k, modulo an additivity relation for L, and the natural action of PSL2n (k), defines a second scissors-congruence group we denote Snq (k). Goncharov proved that the graded vector space S•q (k) is a Hopf algebra, and made a conjecture in [29] which in the number field case states that (1.7)

φq : H 1 (Snq (k)) −→ Ext1MT(k) (Q(0), Q(n))

is an isomorphism. The surjectivity would imply that (1.1) should be generated by quadric motives. We will prove this when k is a totally real field. In this case, the pair (Q, L) can be interpreted as a simplex in hyperbolic n-space. 1.2.3. Polylogarithms. A third approach is via Beilinson and Deligne’s reformulation [5] of Zagier’s conjecture on values of Dedekind zeta functions [28]. For any number field k, one can construct a polylogarithm motive pn (z) ∈ MT(k) parametrized by an element z ∈ k\{0, 1}, whose Hodge realisation is given by the


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polylogarithm mixed Hodge structure. In the case n = 2, this can be written as the matrix:   ∞ 1 0 0 X zk   Li1 (z) 2πi 0 for n ≥ 1 . (1.8) , where Lin (z) = kn Li2 (z) 2πi log z (2πi)2 k=1

The Bloch group Bn (k) is defined by the set of Q-linear combinations of parameters z ∈ k\{0, 1} satisfying an admissibility condition, modulo the set of all functional equations of the nth polylogarithm. Beilinson and Deligne constructed a map: (1.9)

φp : Bn (k) −→ Ext1MT(k) (Q(0), Q(n)) .

The ‘hard part’ of Zagier’s conjecture consists in showing that φp is surjective, i.e., that (1.1) is generated by polylogarithm motives. There is a more precise formulation in terms of polylogarithm complexes due to Goncharov [32]. 1.2.4. Zeta values. The surjectivity of any of the maps (1.5), (1.7), (1.9) would imply a statement about values of zeta functions, as follows. Let ζk (s) denote the Dedekind zeta function of k, and let ζk∗ (1 − n) denote the leading coefficient in the Taylor expansion of ζk (s) at s = 1 − n, for any integer n ≥ 2. The image of the (normalised) Borel regulator (1.10)

rB : K2n−1 (k) ⊗ Q −→ Rn± ,

is a Q-lattice Λn (k) whose covolume is well-defined up to multiplication by an element in Q× . Using the isomorphism of rational K-theory with the stable cohomology of the linear group, Borel proved by an analytic argument in [11] that (1.11)

ζk∗ (1 − n) ∼Q× covol(Λn (k)) .

If one combined this with the isomorphism (1.1), compared the regulators rB and rH 1, and proved the surjectivity of one of the maps φ above, one would obtain a concrete formula for ζk∗ (1 − n) modulo rationals. In the case of the map φh , this would express ζk∗ (1 − n) as a determinant of Aomoto polylogarithms, and in the case of the map φp , this would express ζk∗ (1 − n) as a determinant of classical polylogarithms. The latter is precisely Zagier’s conjecture, which is at present only known in the cases n = 2 and n = 3 [31]. However, such an approach would be highly circuitous, and one should seek a direct motivic proof. In this paper, we do exactly that in the totally real case. We prove that the map φq for quadric scissors-congruence groups is surjective for every totally real number field k, and directly relate the covolume of the image of the Hodge regulator rH with ζk∗ (1 − n), for all n ≥ 2, without using Borel’s result (1.11). This gives a set of generators for (1.1) for any totally real number field in terms of quadric motives. 1.3. Main Results. Let k be a totally real number field. If n is even, the group (1.1) vanishes. For every odd n ≥ 3, we construct a canonical Dedekind zeta motive

(1.12)

motn (k) ∈ MT(k) ,

using arithmetic subgroups of the special orthogonal group SO(2n−1, 1) (see below). The motive motn (k) is a sum of products of quadric motives (1.6) defined over k, and carries a natural framing. 1This is not straightforward: see, for example [17], and the remarks in §1.6 of [22]

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Theorem 1.1. The framed equivalence class of the element motn (k) satisfies [motn (k)] ∈ det Ext1MT(k) (Q(0), Q(n)) , and its image under the regulator (1.3) is a non-zero rational multiple of ζk∗ (1 − n). Theorem 1.1 is a motivic analogue of Borel’s theorem (1.11) in the totally real case. However, the proof of theorem 1.1 is entirely different, as it uses rH rather than rB , and works completely inside the category MT(k). This opens up the possibility of studying other realisations of motn (k), and not just its Hodge realisation. Theorem 1.2. If k is totally real, the map φq of (1.7) is surjective. Thus every element in Ext1MT(k) (Q(0), Q(n)), for k a totally real number field, is obtained from a linear combination of quadric motives. Theorems 1.1 and 1.2 also hold for certain quadratic extensions of k, and in fact for any number field (with no restriction on the complex places) in the particular case n = 2. The element motn (k) is defined in the following way. Let Hm denote hyperbolic space of dimension m, and let Ok denote the ring of integers of k. Then any torsion-free subgroup Γ of finite index of SO(2n − 1, 1)(Ok ) acts properly discretely on r = [k : Q] copies of H2n−1 . The quotient is a product-hyperbolic manifold M = H2n−1 × . . . × H2n−1 /Γ . Then, using a trick due to Zagier, one can show that M can be triangulated using products of hyperbolic geodesic simplices defined over k. By an idea due to Goncharov, a hyperbolic geodesic simplex in the Klein model for H2n−1 is a Euclidean simplex L inside a smooth quadric Q, and so defines a quadric motive (1.6). One shows that the total motive of M , obtained by summing over all products of simplices in the triangulation, gives rise to an element motn (k) ∈ MT(k) whose framed equivalence class The gluing relations between simplices imply that N is well-defined. 1 Ext (Q(0), Q(n)), generalizing a result due to Goncharov [motn (k)] ∈ MT(k) [k:Q] for ordinary hyperbolic manifolds. Theorem 1.1 is then proved using the fact that vol(M ) = α π nr ζk∗ (1 − n) , for some α ∈ Q× , which follows from a Tamagawa number argument. This in turn implies theorem 1.2, using the fact that the rank of Ext1MT(k) (Q(0), Q(n)) is exactly r, which follows from (1.2). There are two corollaries. Corollary 1.3. Let M ∈ Ext1MT(k) (Q(0), Q(n)). The periods of M under π n rH are Q-linear combinations of volumes of hyperbolic (2n − 1)-simplices defined over k. Corollary 1.4. The special value ζk∗ (1 − n) is, up to a rational multiple, a determinant of (π −n -times) volumes of hyperbolic (2n − 1)-simplices defined over k. This result is in the spirit of, but weaker than, Zagier’s conjecture. The proof that the zeta value is a determinant uses the rank calculation (1.2) in an essential way, and does not seem to follow directly from the geometry of the triangulation above. When n = 2, however, the previous results hold for all number fields L. In this case, it is not hard to show that quadric motives and dilogarithm motives coincide, thereby giving another proof of Zagier’s conjecture for n = 2.


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1.4. Plan of the paper. In §2 we introduce some notations and recall some properties of spaces of constant curvature. A product-hyperbolic manifold is defined to be a complete Riemannian manifold of finite volume which is modelled on prodQ ni ucts of hyperbolic spaces N i=1 H . In §3, we list all non-exceptional arithmetic product-hyperbolic manifolds (§3.1), and compute their covolumes up to a rational multiple. A typical example is given by a quadratic form q(x0 , . . . , xn ) = −dx20 + x21 + . . . + x2n ,

where d ∈ k × , for k a totally real field, such that d is positive for t embeddings of k, and negative for the remaining embeddings of k. If Ok denotes the ring of integers of k, then any torsion-free subgroup Γ ≤ SO(q, Ok ) of finite index acts properly discretely on (Hn )t , and defines a product-hyperbolic manifold M = (Hn )t /Γ .

(1.13)

Another family of examples comes from skew-hermitian forms on a quaternion algebra over k, and for n = 2, a third type which are quotients of (H2 )a × (H3 )b . These come from the exceptional isomorphism SO(3, 1) ∼ = PSL2 (C). Theorem 3.5 lists their covolumes which, up to rational factors and powers of π, are: ζk∗ (1 − n) ,

L∗k (χ, 1 − n) ,

and ζL∗ (−1) ,

where Lk (χ, s) = ζk(√d) (s)ζk (s)−1 , for k, d as above, and where L can be any number field. This follows from a well-known Tamagawa number computation. In §4, we explain how to decompose an arbitrary product-hyperbolic manifold M into a finite sum of products of hyperbolic geodesic simplices (1.14)

M=

N X i=1

(i) ∆j

(i)

(i)

∆1 × . . . × ∆N ,

has at most one vertex at infinity. This uses a decomposition of M where into compact and cuspidal parts which is classical when N = 1, but requires strong reduction theory when N > 1. The key idea is an inclusion-exclusion argument originally due to Zagier, and the main result is stated in proposition 4.10 and corollary 4.12. In §5, we begin by recalling some general properties of framed mixed Tate motives, and then show how any product-hyperbolic manifold M defines a motive via its decomposition (1.14). The main idea is due to Goncharov in the case of ordinary hyperbolic manifolds (when N = 1), but our approach differs from his for non-compact manifolds. Let ∆ denote a geodesic simplex in Hm . In the Klein model, the hyperbolic space Hm can be represented as the interior of the unit ball in Rm , and the absolute ∂Hm can be identified with the unit sphere {x21 +. . .+x2m = 1}. Since geodesics are Euclidean lines in this model, the facets of ∆ define m + 1 hyperplanes L0 , . . . , Lm in general position. Viewing Rm as the set of real points of an affine piece of Pm , the unit sphere is given by a smooth quadric Q ⊂ Pm . If the Li are defined over a field k, one can consider the motive [29] Sm (1.15) h(∆) = H m Pm \Q, i=0 Li \(Li ∩ Q) ∈ MT(k) . If m is even, h(∆) has no non-trivial framing, but if m = 2n − 1 is odd, it is framed by the class corresponding to the volume form on Hm , and by the class corresponding to the hyperbolic simplex ∆. If the simplex has a vertex at infinity

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x, the divisor (Q, L) will no longer be normal crossing and we must blow up the point x. The corresponding motive is now defined to be e m \Q, e Sm L e i \(L e i ∩ Q) e ∈ MT(k) , (1.16) h(∆) = H m P i=−1

e m is the blow-up of Pm at x, L e −1 is the exceptional divisor, and Q, e L e i are the where P strict transforms of Q, Li . The framings are defined as before, and in both cases we denote the framed motive by mot(∆). There is, however, an interesting difference between the motives (1.15) and (1.16) which is due to the fact that certain periods of (1.15) are given by the hyperbolic length of the edges of ∆, and these can be infinite if a vertex of ∆ is at infinity. In §5.3, explicit examples are calculated in dimensions 1 to 3 which are sufficient to illustrate the general pattern. We then define the motive of a product-hyperbolic manifold M modelled on QN products of odd-dimensional hyperbolic spaces i=1 H2ni −1 to be

(1.17)

mot(M ) =

N X i=1

(i)

(i)

mot(∆1 ) ⊗ . . . ⊗ mot(∆N ) .

That mot(M ) does not depend on the particular decomposition (1.14), follows from a subdivision lemma 5.11, which states that m X mot(∆(x0 , . . . , xi−1 , y, xi , . . . , xm )) , (1.18) mot(∆(x0 , . . . , xm )) = i=0

where x0 , . . . , xm , y are points in Hm , at most one of which is on the absolute, and ∆(a0 , . . . , am ) denotes the geodesic simplex with vertices at a0 , . . . , am . We then prove in §5.2.4, using only (1.18) and some simple properties of h(∆) that (1.19)

mot(M ) ∈

N O

Ext1MT(Q) (Q(0), Q(ni )) .

i=1

This is equivalent to showing that mot(M ) lies in the kernel of the reduced coproduct on the Hopf algebra of framed equivalence classes of mixed Tate motives. The idea is that the image of this coproduct can be written as a sum over all faces in the decomposition (1.14). Since this decomposition lifts locally to give a triangulaQN tion of i=1 Hni , the coefficient of any given face F in the image of the coproduct must vanish, since (1.18) implies that we can modify the triangulation in such a way that F does not even appear. A different way to prove this is by following Goncharov’s analytic approach using generalised Dehn invariants for products of hyperbolic spaces. This is done in §5.2.9, using results of §4.6 and §4.7, which are not used anywhere else and can be skipped. In the case where M is defined arithmetically, for instance by (1.13), we then study a certain twisted Galois action on M to prove that the element mot(M ) is a determinant. The main result of the section is theorem 5.20. In §6, we combine the various elements to prove the main theorems of this paper, which generalise theorems 1.1, 1.2 and their corollaries. The corresponding results in the exceptional case n = 2 are given in §6.3, and some open questions and possible directions for future research are discussed in §6.4. Finally, in §7 we compute a very simple and explicit example of an L(χ, 3)-motive, based on a remarkable computation due Its period is the L-value L(χ, 3) = ζk′ (3)ζk−1 (3), √ to Bugaenko. ′ where k = Q( 5) and k is a quadratic extension of k.


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2. Spaces of constant curvature. As is well-known, a homogeneous space of constant curvature is of one of three types: Euclidean space, spherical space, and hyperbolic space, which have constant sectional curvature 0,+1, and -1, respectively. We use the following notations: • For n ≥ 1, Euclidean space En is Rn equipped with the scalar product (x, y) = x1 y1 + . . . + xn yn . We denote its group of symmetries E(n) = T(n) ⋊ O(n), where T(n) ∼ = Rn is the group of affine translations, and O(n) is the orthogonal group. • Let n ≥ 2, and let x0 , . . . , xn denote coordinates in En+1 . Spherical space Sn = {x ∈ En+1 : x20 + . . . + x2n = 1} inherits the metric ds2 = dx20 +. . .+dx2n , and has symmetry group O(n+1). • Let n ≥ 2, and let Rn,1 denote Rn+1 equipped with the inner product (x, y) = −x0 y0 + x1 y1 + . . . + xn yn . Hyperbolic space Hn is defined to be the half-hyperboloid: Hn = {x ∈ Rn,1 : (x, x) = −1, x0 > 0} . Let SO(n, 1) denote the group of matrices preserving this scalar product. It has two components. Let SO+ (n, 1) denote the connected component of the identity, which is the group of orientation-preserving symmetries of Hn . The invariant metric on Hn is given by ds2 = −dx20 + dx21 + . . . + dx2n . We will consider complete manifolds M which are locally modelled on products of spaces of constant curvature, i.e., Riemannian products of the form Y Y Y Xn = Hni × Snk , Enj × i∈Ih

j∈Ie

k∈Is

where n = {ni : i ∈ I}, and I = Ih ∪ Ie ∪ Is . We will use a Roman superscript to denote a single space of constant curvature Xn ∈ {En , Sn , Hn }, and a gothic superscript to denote a product as above. The space Xn has a canonical volume form which we denote by dv. A complete, orientable manifold modelled on Xn of finite volume is of the form M n = Xn /Γ, where Γ is a discrete torsion-free subgroup of the group of motions of Xn . We are mainly interested in the quantity vol(M n ) mod Q× , which only depends on the commensurability class of Γ. Since the orthogonal group is compact, any discrete group acting on a product of spheres is finite, and thus torsion, so we can always assume that there are no spherical components. In this case, we will say that M n is a flat-hyperbolic manifold. If it is modelled only on products of hyperbolic spaces, it will be called product-hyperbolic. The following theorem was proved in 1984 by Margulis. Theorem 2.1. [43] Let G be a connected semisimple linear algebraic group without compact factors. If the real rank of G is strictly greater than 1 then every irreducible lattice in G is arithmetic.

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P The real rank of Xn is |Ih | + i∈Ie ni , since it coincides with the dimension of a maximal flat geodesic subspace. The theorem therefore implies that any irreducible discrete group of finite covolume Γ acting on Xn is either arithmetic, or else Xn = Hn and Hn /Γ is an ordinary hyperbolic manifold. It follows that M n is commensurable to a product of manifolds of the following type: • a sphere Sn , whose volume is a rational multiple of a power of π, • a torus En /Λ, where Λ ∼ = Zn , whose Q volume can be any real number, • a product-hyperbolic manifold i Hni /Γ, where Γ is an arithmetic group of motions, • an ordinary hyperbolic manifold Hn /Γ where Γ is non-arithmetic. Therefore only the volumes of product-hyperbolic manifolds are of real numbertheoretic interest, and likewise, these are the manifolds which have interesting motives. In §3 we compute the volumes modulo Q× of all non-exceptional arithmetic product-hyperbolic manifolds in terms of zeta functions and L-functions of number fields. The case of non-arithmetic manifolds is more mysterious, since there does not appear to exist a suitable volume formula for them at present. 2.1. Models of hyperbolic spaces and the absolute. In addition to the hyperboloid (vector) model of hyperbolic space defined above, we will need to consider the following models ([1], I,§2). 2.1.1. The Poincaré upper-half space model. Let e0 = (−1, 0, . . . , 0) ∈ Rn , and consider the map (x0 , . . . , xn ) 7→ (x′1 , . . . , x′n ), where x′i = xi /(1 + x0 ). This maps the hyperboloid model of Hn to the Poincaré unit ball. Then consider the map (x′ + e0 ) x′ 7→ 2 ′ − e0 ||x + e0 ||2 ′2 where x′ = (x′1 , . . . , x′n ) and ||x′ + e0 ||2 = (x′1 − 1)2 + x′2 2 + . . . + xn . Composing n the two gives an isometry from H to the Poincaré upper-half space:

Un = {(z1 , . . . , zn−1 , t) : (z1 , . . . , zn−1 ) ∈ Rn−1 , t > 0} , Pn−1 2 with the metric t−2 ( i=1 dzi + dt2 ). The absolute ∂Hn is identified with the Eun clidean plane Rn−1 × {0} ⊂ U at height t = 0, compactified by adding the single point at infinity ∞. Geodesics in this model are vertical line segments (which go to ∞) or segments of circles which meet the absolute at right angles. In this model, hyperbolic angles coincide with Euclidean angles. (2.1)

2.1.2. The Klein (projective) model. The map (x0 , . . . , xn ) 7→ (y1 , . . . , yn ), where yi = xi /x0 , gives an isomorphism of the hyperboloid model with the Klein model: n X yi2 < 1} (2.2) Kn = {(y1 , . . . , yn ) : r2 = i=1

Pn Pn equipped with the metric ds2 = (1 − r2 )−2 [(1 − r2 ) i=1 dyi2 + ( i=1 yi dyi )2 ]. The action of SO+ (n, 1) on Kn is by projective transformations, and extends continuously to the boundary ∂Kn . Geodesics in this model are ordinary Euclidean hyperplanes, but hyperbolic and Euclidean angles do not agree in this case.


11

3. Volumes of product-hyperbolic manifolds and L-functions We list all discrete arithmetic groups which act on products of hyperbolic spaces, and compute their covolumes up to a rational multiple. By a Tamagawa number argument, the covolumes are expressed as values of L-functions. 3.1. Arithmetic groups acting on product-hyperbolic space. There are four basic types of arithmetic groups which act on products of hyperbolic spaces. Type (I) : Let k/Q be a totally real field of degree r, and let O denote an order in k. Let 1 ≤ t ≤ r. Consider a non-degenerate quadratic form (3.1)

q(x1 , . . . , xn ) =

n X

aij xi xj

aij ∈ k , aij = aji ,

where

i,j=0

Pn σ and suppose that σq = i,j=0 aij xi xj has signature (n, 1) for t infinite places σ ∈ {σ1 , . . . , σt } of k, and is positive definite for the remaining r − t infinite places of k. Let SO+ (q, O) be the group of linear transformations with coefficients in O preserving q, which maps each connected component of {x ∈ Rn+1 : σiq(x) < 0} to itself for 1 ≤ i ≤ t. Let Γ be a torsion-free subgroup of SO+ (q, O) of finite index. Q It acts properly discretely on ti=1 Hn via the map Γ ֒→

t Y

SO+ (n, 1)(R)

i=1

A

7→

( σ1A, . . . ,

σt

A) .

Type (II) : Suppose that n = 2m − 1 is odd. Let k/Q be a totally real number field of degree r, and let 1 ≤ t ≤ r. Consider a quaternion algebra D over k such that Dσ = D ⊗k,σ R is isomorphic to M2×2 (R) for all embeddings σ of k. Let (3.2)

Q(x, y) =

m X

xi aij yj

where

i,j=1

aij ∈ D, aij = −aji ,

be a non-degenerate skew-Hermitian form on Dm , where x 7→ x denotes the conjugation map on D, which is an anti-homomorphism. For each embedding σ of k, Pm the signature of σQ, where σQ(x, y) = i,j=1 xi σaij yj , is defined as follows. Let Dσ ∼ = M2×2 (R) = R ⊕ iR ⊕ jR ⊕ kR, where i2 = j2 = 1 and ij = −ji = k. The endomorphism of Dσ given by right multiplication by i is of order two and has eigenvalues ±1. Let Dσ,± = {x ∈ D : xi = ±x} .

m m m m ⊕ Dσ,− , and because one has Dσ,− = Dσ,+ j, the diIt follows that Dσm = Dσ,+ m mension of Dσ,+ is 2m. Writing x = x+ + x− , where x+ i = x+ and x− i = −x− , one verifies that σQ(x+ , y+ ) = σQ(x+ i, y+ ) = −i σQ(x+ , y+ ), and σQ(x+ , y+ ) = σ Q(x+ , y+ i) = σQ(x+ , y+ )i. It follows that σ σ

Q(x+ , y+ ) = fσ (x+ , y+ )(j − k) ,

where fσ (x+ , y+ ) ∈ k .

Since Q is skew-hermitian, fσ is a non-degenerate symmetric bilinear form. The signature of σQ is defined to be the signature of fσ . Let φσ denote the map which to Q associates the bilinear form fσ . The form fσ uniquely determines σQ, for instance, via the formula −j σQ(x− , y+ ) = σQ(x− j, y+ ) = fσ (x− j, y+ )(j − k).

12

FRANCIS C.S. BROWN

Suppose, therefore, that σQ has signature (2m − 1, 1) for t embeddings σ ∈ {σ1 , . . . , σt } of k, and is positive definite for the remaining r − t embeddings. Let O be an order in D, and let U+ (Q, O) denote the group of O-valued points of the unitary group which preserves the connected components of {x ∈ σiD : Q(x, x) < 0} for 1 ≤ i ≤ t. If Γ is a torsion-free subgroup of U+ (Q, O), then it acts properly Qt discretely on i=1 Hn via the map (φσ1 , . . . , φσt ) : Γ ֒→ SO+ (n, 1)t (R).

Type (III) : Let L/Q denote a number field of degree n with r1 real places and r2 ≥ 1 complex places, and let 0 ≤ t ≤ r1 . Let B denote a quaternion algebra over L which is unramified at t real places, and ramified at the other r1 − t real places Q of L. Then v|∞ (B ⊗L Lv )∗ = (H∗ )r1 −t × GL2 (R)t × GL2 (C)r2 , where H denotes Hamilton’s quaternions. Let O denote an order in B and let Γ be a torsion-free subgroup of finite index of the group of elements in B of reduced norm 1. Then Γ defines a discrete subgroup of PSL2 (R)t × PSL2 (C)r2 and acts properly discontinuously on (H2 )t × (H3 )r2 (see [20, 13].) Type (IV) : There are further exceptional cases for H7 which are related to Cayley’s octonions. These will not be considered here. Remark 3.1. In the case when n is odd, every arithmetic group of type (I) can also be expressed as an arithmetic group of type (II). Thus in type (I) one can assume that n is even. Proposition 3.2. Let Γ be an irreducible arithmetic group acting on a product of hyperbolic spaces. Then Γ is commensurable to one of the four types listed above.

Proof. Let G be an algebraic group defined over a number field. By restricting scalars, we can assume that G is defined over Q. The discrete subgroup G(Z) then acts on the symmetric space G/K(R), where K is a maximal compact subgroup of G, and we will give conditions on G to ensure that this is isomorphic to a product of hyperbolic spaces. Since we have assumed G(Z) to be irreducible, G must be Q-simple. It follows that G is a restriction of scalars, for let H denote a Q-simple factor of G. The product of the elements in the orbit of H under Gal(Q/Q) defines a Q-subgroup of G, which is therefore isogenous to G since it is Q-simple. Thus G = Rk/Q H for some number field k. Q In this case, G(R) ∼ = v|∞ H(kv ), where kv denotes the completion of k at an infinite place v. In order to be a hyperbolic space form, G(R) must have at least one simple factor isogenous to SO(n, 1). First we suppose that one such factor occurs at a complex place v. If h denotes the Lie algebra of H, then we have hC ∼ = so(n, 1) and by complexifying, hC ⊕hC ∼ = so(n+1)C . But so(n+1)C is simple for all n except if n = 3. In this case, there is indeed an exceptional isomorphism sl(2)C ∼ = so(3, 1), so H must be a form of SL2 . Therefore in each real place v, H(kv ) is a real form of SL2 which can be of two types: SL2 (R) ∼ = SO+ (2, 1)(R), or SO(3)(R), which is compact. It follows that for any form H of SL2 , the corresponding symmetric space is a product of hyperbolic spaces H2 and H3 . Now SL2 has no non-trivial outer forms, and its inner forms are classified by H 1 (Gal(Q/Q), PSL2 ). The same group classifies quaternion algebras, and one deduces that G is the group of elements of reduced norm one in a quaternion algebra B [56]. This gives case (3). Now suppose that H(kv ) is isogenous to SO(n, 1) at some real place v. It follows that for each infinite place v of k, H(kv ) is either SO(n + 1)(C) if v is complex, or a


13

real form of SO(n + 1)(C) if v is real. If v is a complex place, the symmetric space SO(n + 1)C /SO(n + 1) can be a product of hyperbolic spaces only if n = 2, and we have already dealt with this case. Thus we can assume from now on that k is totally real. The real forms of so(n + 1)C are either so(p, n + 1 − p), for 1 ≤ p ≤ n + 1 − p, or u∗m (H) when n + 1 = 2m, and where H denotes Hamilton’s quaternions ([35], pp 138,229). The former has real rank p, and the latter has real rank ⌊m/2⌋. Using, for example, the fact that hyperbolic spaces have rank one, and using the classification of real semi-simple Lie algebras ([35], pg 135) to rule out the cases u∗3 (H), u∗2 (H), we deduce that at each infinite place v, hkv must be either so(n, 1) or so(n + 1). The orthogonal group SO(n+1) has exactly one outer form when n = 2m is even, since its Dynkin diagram is of type Bm and has no symmetries. If n = 2m−1 ≥ 5 is not equal to 7, its Dynkin diagram is of type Dm which has a non-trivial symmetry, so there are exactly 2 outer forms. When n = 7, its Dynkin diagram is of type D4 , whose symmetry group is the symmetric group on 3 letters. We will exclude the non-classical cases coming from the triality forms of D4 (type (IV)). Suppose first that n is even, and therefore H is an inner form of SO(n + 1). It is well-known from the classification of arithmetic groups ([64], [48], §2.3) that H is isogenous to SO+ (q) for some quadratic form q defined over k. By the above, the condition that the symmetric space of H be product-hyperbolic is that the signature of q is (n, 1) for at least one real embedding of k, and positive definite for all remaining embeddings. This gives type (I), for even n. Finally, suppose that n = 2m − 1 ≥ 5 is odd, and that H is an outer form of SO(n + 1). The classification of arithmetic groups ([48], proposition 2.20) states that H is given by the unitary group of a skew-symmetric Hermitian form Q of dimension m over a quaternion algebra D over k. Over each infinite place v, D is either isomorphic to Hamilton’s quaternions H, or the matrix algebra M2 (R). By the above, the former case gives rise to the symmetric space of u∗m (H) which is not hyperbolic and so must be ruled out. Hence D must split over all real embeddings of k. The signature of Q must be (n, 1) for at least one embedding of k, and Q must be positive definite for the remaining embeddings. This gives type (II). This also subsumes the cases of type (I) for odd n. Remark 3.3. The main results of this paper are obtained in the simplest case, where q is given by the trivial quadratic from −x20 + x21 . . . + x2n . Then we obtain the standard orthogonal group SO(n, 1), and Γ is just any torsion-free subgroup of finite index of Rk/Q SO+ (n, 1)(Z) = SO+ (n, 1)(Ok ). In the case n = 2, an important example is given by B = M2 (L), where L is a number field as in (III). Then Γ is just a torsion-free subgroup of finite index of the Bianchi group PSL2 (OL ).

14

FRANCIS C.S. BROWN

3.2. Tamagawa numbers and volumes. We recall the definition of Tamagawa numbers, and give an explicit formula for the covolume of an arithmetic group modulo Q× , following Ono, and avoiding all use of Bruhat-Tits theory. We compute the terms in this formula in the case of product-hyperbolic manifolds (theorem 3.5, corollary 3.10). In principle one can determine the exact volumes of arithmetic quotients of semi-simple groups using the work of Prasad [49] (see also [4, 20, 34]), but this is not required here. 3.2.1. Tamagawa numbers. Let G be an affine algebraic group defined over a number field k. For each place v of k, let kv denote the completion of k with respect to v, Ov its ring of integers, and Fv its residue field. Let Ak denote the group of adèles of k, and let G(Ak ) denote the group of Ak -valued points of G [63]. Since G is affine, we can fix an embedding G → GLn and simply take G(Ak ) = G∩GLn (Ak ). Then G(Ak ) is the restricted direct product: Y Y G(Ov ) , G(Ak ) = lim G(kv ) × → S

v∈S

v ∈S /

where the direct limit is over all finite sets of places S of k, ordered by inclusion. The group G(Ak ) is locally compact, and therefore has a left Haar measure which is defined as follows [63]. Let ω be a left-invariant non-zero volume form on G which is defined over k. For all x ∈ G, we may write ω(x) = f (x) dx1 . . . dxn ,

where x = (x1 , . . . , xn ) are local affine coordinates for G, and f is a k-rational function in the xi . For all places v of k, there is a local measure ωv on Gkv : (3.3)

ωv (x) = |f (x)|v (dx1 )v . . . (dxn )v ,

R where (dx1 )v are Haar measures on kv (normalised so that Ov (dxi )v = 1 when v is finite, and ordinary Lebesgue measure when v is infinite), and |.|v is the module of kv 2. We will now assume that G is semi-simple. The Tamagawa measure of G is the measure induced on G(Ak ) by the product over all places v of k: Y Ω = |dk |− dim G/2 ωv , v

where dk is the absolute discriminant of k. One shows that Q since G is semi-simple, the product converges absolutely for the adèle topology, i.e., v 0. Let gk denote the Lie algebra of G over k. One can define invariant volume forms on Xv , K(kv ) and G(kv ), which we denote by dXv , dKv and dGv , respectively, which are canonical up to a sign. Let B denote the Killing form on gk , and let {U1 , . . . , Ud } be a k-basis of gk . Suppose that this basis contains as a subset a basis {V1 , . . . , Ve } of K, and let {Vi′ } ⊂ {Ui′ } denote the corresponding dual bases of invariant 1-forms. Following ([45], §3.2) we set 1/2 (3.4) dGkv = det B(Ui , Uj ) v U1′ ∧ . . . ∧ Ud′ v , 1/2 dKkv = det B|K (Vi , Vj ) v V1′ ∧ . . . ∧ Ve′ v ,

which define left-invariant volume forms, and do not depend on the choices of basis. Let dXv be the unique invariant volume form on Xv satisfying dXv ∧d Kkv = d Gkv . Q The canonical volume form on X is defined to be dX = v|∞ dXv , where the signs are chosen so that the form is positive. In order to express the covolume of an arithmetic subgroup acting on X, one defines two invariants of G which are well-defined up to multiplication by Q× . The first is a transcendental invariant, defined as follows. For each finite place v of k, let qv denote the number of elements in the residue field Fv at v. Let #G(Fv ) be the number of points of G over Fv . The L-value of G is the following product over all finite places of k: Y q dim G v (mod Q× ) . (3.5) LG = #G(F ) v v0 : t ≥ r}.

Its boundary, the horosphere, is isometric to a Euclidean plane En−1 at height Q ni n r. Now let X = i∈I H . For any subset S ⊂ I, and any set of parameters r = {ri > 0}i∈S indexed by S, we define the corresponding product-horoball to be Y Y BS (r) = Hni ⊂ Xn . Bni (ri ) × i∈S

i∈I\S

Q Q Its horosphere is i∈S ∂Bni (ri ) × i∈I\S Hni which is diffeomorphic to a flatQ Q hyperbolic space i∈S Eni −1 × i∈I\S Hni . The cusps of a product-hyperbolic mani|S|

fold are isometric to a product F ×R>0 , equipped with the metric e−2(t1 +...+t|S| ) ds2 + dt21 + . . . + dt2|S| , where F is a flat-hyperbolic manifold with metric ds2 , and ti is |S|

the coordinate on the ith component of R>0 .

Theorem 4.1. Any complete product-hyperbolic manifold M of finite volume has a decomposition into disjoint pieces M=

E [

M(ℓ)

ℓ=0

where each piece M(ℓ) is isometric to a cusp, i.e., M(ℓ) ∼ = Fℓ × Rk>0 , where k ≥ 0 and Fℓ is a compact flat-hyperbolic manifold with boundary. The proof of the theorem is different in the case when M is arithmetic or nonarithmetic. By theorem 2.1, an irreducible non-arithmetic lattice is necessarily of rank one and acts on a single hyperbolic space. In this case, M is an ordinary hyperbolic manifold of finite volume and the decomposition is well-known. The key result is the decomposition of a hyperbolic manifold into thick and thin parts using the Gromov-Margulis lemma ([2], III, §10 and appendix. See also [15, 25]). In the case of arithmetically-defined groups, the decomposition follows from strong reduction theory, which is outlined below. There does not appear to be a result which covers both cases at the same time. 4.1.1. Rank ≥ 2: Precise reduction theory. Let G denote any connected semi-simple algebraic group defined over Q, and let Γ ⊂ GQ be an arithmetic subgroup. Let K denote a maximal compact subgroup of G, and let X = K\G. Classical reduction theory provides the existence of a fundamental set Ω ⊂ X for Γ such that ΩΓ = X, and such that for all g ∈ G(Q), the set {γ ∈ Γ : gΩ ∩ γΩ 6= ∅} is finite. In order to obtain a decomposition of M = X/Γ into disjoint ends, one needs to construct a fundamental domain for Γ, which requires a more precise version of reduction

22

FRANCIS C.S. BROWN

theory [42]. Let P denote any Q-parabolic subgroup of G. Let NP denote the unipotent radical of P , and let AP denote the identity component of the maximal Q-split torus in the center of P/NP . The adjoint action of (a lift of) AP gives a decomposition of the Lie algebra of NP into rational root spaces. The parabolic group P corresponds to a Weyl chamber in this rational root system. This chamber is determined by a set RP of positive simple roots. There is an analytic isomorphism ∼

AP × e(P ) −→ X , given by the geodesic action of AP on the horosphere e(P ) = X/AP . Let ΓP denote the stabiliser of P in Γ. It follows from a theorem due to Borel that there are only finitely many Γ-conjugacy classes of maximal Q-parabolic subgroups Q1 , . . . , Qm ⊂ G . Let ai denote the Lie algebra of AQi for 1 ≤ i ≤ m, and define a=

m M

ai .

i=1

This is a space of parameters measuring the distance out to infinity in each of the ends of M . Since a parabolic Q-subgroup may contain further such subgroups, such parameters must be chosen in a compatible manner in order to make sense. One defines a map IP : a → aP for any Q-parabolic subgroup P of G to take these compatibility conditions into account. Let AP,T = {exp(H) ∈ AP |α(H) > α(IP (T )), for all α ∈ RP } . The following theorem is clearly explained in [42], see also [55]. Theorem 4.2. Let P0 = G, and P1 , . . . , PE be representatives of Γ-conjugacy classes of rational parabolic subgroups of G. Then for any T ∈ a sufficiently large in each coordinate, there exist bounded sets ωi ⊂ e(Pi ) such that (1) Each set ωi APi ,T is mapped injectively into M = X/Γ. (2) The image of ωi in e(Pi )/ΓPi is compact. (3) There is a decomposition of M as a disjoint union M=

E a

Mi

i=0

where Mi = ωi APi ,T /Γ for 0 ≤ i ≤ E. Q ni In the case which interests us, X = H is a product of hyperbolic spaces, which have rank 1. The root systems we obtain are therefore orthogonal products of one-dimensional root systems, and the parabolic subgroups of G stabilise a set of components at infinity. By the discussion at the beginning of §4.1, it follows that e(Pi ) is a horosphere, and ΓPi is a subgroup of the groupQof isometries Q of flathyperbolic space. In a product of upper-half space models Uni = i {(zi , ti ) : zi ∈ Rni −1 , ti ∈ R>0 }, the distance functions in aP are given by a subset of heights {ti }. Each set Mi is therefore a cone over a compact flat-hyperbolic manifold ωi /ΓPi . This implies the decomposition theorem 4.1 in the arithmetic case. 4.2. Geodesic simplices and rational points. We must triangulate a producthyperbolic manifold M with simplices whose vertices have coordinates in a number field, and study a twisted action of the symmetric group on this triangulation.


23

4.2.1. Rational points on product-hyperbolic manifolds. Let k be a subfield of R, and let X = K\G be a space of constant curvature (§2). The set of k-rational points Xk = K(k)\G(k) is the G(k)-orbit of the point x0 ∈ X stabilised by K. For Euclidean and spherical spaces En and Sn , we can take the set of points with k-rational coordinates. The same is true for Hn in the upper-half space, and hyperboloid models. However, in the Klein unit ball model, Hnk is given by: X yi2 ∈ k ×2 } . {(y1 , . . . , yn ) ∈ k n such that 1 − i

Hnk

∂Hnk

n

n

The sets and are dense in H and ∂H respectively, and we will write QN n Hk = Hk ∪ ∂Hnk . Now consider products of hyperbolic spaces Xn = i=1 Hni . Let S = (k1 , . . . , kN ) denote an ordered set of fields, with ki ⊂ R. We define the set of Q ni S-rational points on Xn to be XnS = N i=1 Hki . We say that a product-hyperbolic n manifold M = X /Γ is defined over S if Γ is conjugate to Γ′ which satisfies Y Γ′ ≤ SO+ (ni , 1)(ki ) . 1≤i≤N

Remark 4.3. In dimension n = 3, there is an exceptional isomorphism between SO+ (3, 1)(R) and PSL2 (C). For any field L ⊂ C, we define the set of L-points H3L of H3 to be the orbit of a fixed rational point x0 under PSL2 (L). In what follows, the fields ki corresponding to hyperbolic components of X of dimension 3 will therefore be viewed as subfields of C rather than R. Theorem 4.4. Let M be a complete product-hyperbolic manifold of finite volume. Then M is defined over S = (k1 , . . . , kr ) where ki are number fields. Proof. For arithmetic groups this follows from the definition. For non-arithmetic groups, it suffices to consider discrete subgroups of SO+ (n, 1), by theorem 2.1. By Weil’s rigidity theorem, Γ is conjugate to a subgroup Γ′ of SO+ (n, 1)(Q). Since Γ′ is finitely generated, its entries lie in a field k ⊂ R which is finite over Q. We define the set of S-rational points of such a manifold M to be (4.1)

MS = XnS /Γ′ .

The set of S-rational points MS is dense in M . Lemma 4.5. The fixed point z ∈ ∂Hn of any parabolic motion γ ∈ SO+ (n, 1)(k) is defined over k, i.e., z ∈ ∂Hnk . Proof. In the Klein model for hyperbolic space, the group SO+ (n, acts by P1)(k) 2 z = 1}. It projective transformations on the absolute ∂Hn = {z1 , . . . , zn : i i follows that a point z ∈ ∂Hn which is stabilised by γ ∈ SO+ (n, 1)(k) satisfies an equation γz = z, which is algebraic in the coordinates of z, and has coefficients in k. If γ is parabolic, then this equation has a unique solution on the absolute. By uniqueness, z coincides with its conjugates under Gal(k/k), and hence z ∈ ∂Hnk . 4.2.2. The twisted action of the symmetric group. Now let Σ = {σ1 , . . . , σN } denote a set of distinct real embeddings of a fixed field k as above. Set ki = σi k, and let S = (k1 , . . . , kN ). For each pair of indices 1 ≤ i, j ≤ N , there is a bijection n n ∼ σi σj−1 : Hkj −→ Hki . Let SN denote the symmetric group on N letters {1, . . . , N }.

24

FRANCIS C.S. BROWN

Definition 4.6. If the dimensions of all hyperbolic components ni are equal, the QN n n symmetric group SN acts on XS = i=1 Hki as follows: −1 −1 π(x1 , . . . , xN ) = (σ1 σπ(1) xπ(1) , . . . , σN σπ(N ) xπ(N ) ) ,

where π ∈ SN .

n

We define the equivariant points of XS to be the fixed points under this action. n This is identified with Hk via the embedding e : x 7→ (σ1 (x), . . . , σN (x)). QN Definition 4.7. A product-hyperbolic manifold M = i=1 Hni /Γ is equivariant with respect to SN if Γ lies in the image of Y e : SO(n, 1)(k) −→ SO(n, 1)(σi (k)) 1≤i≤N

A

7→

(σ1 (A), . . . , σN (A)) .

Note that the fixed point of a parabolic motion on an equivariant product-hyperbolic manifold is necessarily equivariant. 4.2.3. Geodesic simplices and the action of SN . Let X be a space of constant curvature of dimension n ≥ 2. If x0 , . . . , xn are n+1 distinct points in X, let ∆(x0 , . . . , xn ) denote the geodesic simplex whose vertices are x0 , . . . , xn . This is defined to be the convex hull of the points {x0 , . . . , xn }. For this to make sense in spherical space, we have to assume that all vertices lie in the same half-sphere. If the points x0 , . . . , xn lie in a geodesic subspace, then ∆(x0 , . . . , xn ) will be degenerate. In the case where X is hyperbolic space Hn , we can allow some or all of the vertices xi to lie on the boundary ∂Hn . One can show that such a simplex ∆(x0 , . . . , xn ) always has finite, and in fact bounded, volume. The simplex ∆(x0 , . . . , xn ) is said to be defined over a field k ⊂ R, if x0 , . . . , xn ∈ Xk . Suppose that we are given a map of fields σ : k ֒→ k ′ . It induces an action on geodesic simplices over defined over k: (4.2)

σ∆(x0 , . . . , xn ) = ∆(σx0 , . . . , σxn ) ⊂ Xk′ .

This action does not extend continuously to the interior of the simplex, and two simplices with non-empty intersection can map to simplices which are disjoint. In a product of spaces of constant curvature, we consider products of the form (4.3)

∆ = ∆1 × . . . × ∆N ,

where ∆i are geodesic simplices in each component. We call this a geodesic productsimplex. It is defined over the fields S = (k1 , . . . , kN ) if ∆i is defined over ki for all 1 ≤ i ≤ N . In the equivariant case S = (σ1 k, . . . , σN k), and all ni are equal, the symmetric group SN acts on geodesic product simplices defined over S as follows: −1 −N (4.4) π(∆1 × . . . × ∆N ) = σ1 σπ(1) ∆π(1) × . . . × σ1 σπ(N ) ∆π(N ) for any π ∈ SN .

4.3. Generalities on virtual triangulations. Let M denote a flat-hyperbolic manifold. In order to decompose M into products of geodesic simplices, we must consider geodesic polytopes in products of Euclidean and hyperbolic spaces, which may have vertices at infinity. Let P denote an n-dimensional convex polytope in Euclidean or hyperbolic space. A faceting F (P ) of P is a finite collection of closed, (n − 1)-dimensional convex polytopes Fi ⊂ ∂P such that: (1) Each face of P is a disjoint union of facets. (2) Any two facets are either disjoint or meet along their common boundaries.


25

The set of all codimension 1 faces of P , for example, defines a faceting of P . A n product-polytope in X is a product of convex geodesic polytopes, and one defines a faceting in a similar manner. For any product-polytope P , let P f denote the polytope P with all infinite components removed. Now let R denote a finite set n of geodesic product-polytopes in X , and suppose we are given a faceting for each product-polytope in R. A set of gluing relations for R is a way to identify all facets of all polytopes P ∈ R in pairs which are isometric. Let a T = Pf/ ∼ P ∈R

denote the topological space obtained by identifying glued facets. Now let x be any point in a facet of P ∈ R. The cycle of x is the set of all polytopes which contain x after gluing. The solid angle at x is defined as follows. Let Y Y Y Y Bx (ε) = Bxj (ε) ⊂ Bxi (ε) × Hnj , Eni × i∈Ie

i∈Ie

j∈Ih

j∈Ih

Q Q denote a product of Euclidean and hyperbolic spheres centered atQx = xi × Q xj of radius ε ≪ 1 in each direction. For every product polytope P = i∈Ie Pi × j∈Ih Pj in the cycle of x, the solid angle at x is Y vol(Pi ∩ Bx (ε)) Y vol(Pi ∩ Bx (ε)) i i × ∈R. θP (x) = vol(Bxi (ε)) vol(Bxi (ε)) j∈Ih

i∈Ie

R is said to be proper if, for all x ∈ R, the sum of the solid angles over all polytopes in the cycle of x sum to 1: X θP (x) = 1 . P

When this is the case, it is possible to glue all the spherical sectors Bx (ε) ∩ P back together to reconstitute Bx (ε). This defines a local chart at x. One proves ([50], §11.1) that T is a flat-hyperbolic manifold if and only if it is proper. It will not necessarily be complete, however. A product-tiling of M is then an isometry (4.5)

f : T −→ M .

Since facets may be strictly contained in the faces of each geodesic simplex, the tiling is not always a triangulation and may resemble a wall of bricks (figure 1).

Figure 1. A triangulation in H2 (left), a product-tiling in E2 (middle), and a virtual product-tiling with crease over E2 (right).

26

FRANCIS C.S. BROWN

We will also need to consider virtual tilings. To define a virtual tiling, consider a continuous surjective map f : T −→ M, of finite degree which is not necessarily étale. We assume that T is proper, and that the restriction of f to the interior of each product polytope P in T is an isometry. We define the local multiplicity of f on P to be +1 if f |P is orientation-preserving, and −1 if f is orientationreversing. The condition that f be a virtual tiling is that the total multiplicity of f is almost everywhere equal to 1. In the case when M is defined over the fields S = (k1 , . . . , kN ), we will say that the (virtual) product tiling is defined over S if the geodesic product simplices which occur in T are defined over S. 4.4. Tiling of product-hyperbolic manifolds. One can construct a producttiling of any complete, orientable, finite-volume flat-hyperbolic manifold M n . In order to do this, we apply a modified version of an inclusion-exclusion argument due to Zagier [65]. We first need the following excision lemma. Lemma 4.8. Let X1 , . . . , Xn and Y1 , . . . , Yn denote any sets, and let ǫ ∈ {0, 1}n. We write ( Xi ∩ Yi if ǫi = 0 , Xi (ǫ) = Xi \(Xi ∩ Yi ) if ǫi = 1 ,

and define Yi (ǫ) similarly. Any union of products (X1 × . . . × Xn ) ∪ (Y1 × . . . × Yn ) can be written X X (X1 ∩Y1 )×. . .×(Xn ∩Yn )+ X1 (ǫ)×. . .×Xn (ǫ)+ Y1 (ǫ)×. . .×Yn (ǫ), 06=ǫ∈{0,1}n

06=ǫ∈{0,1}n

where a plus sign denotes disjoint union.

X X1 (1)×X2 (1)

X1 (0)×X2 (1)

111111111 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 X (1)×X (0)111111111 X ∩Y Y (1)×Y (0) 000000000 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 000000000 111111111 1

2

1

Y1 (0)×Y2 (1)

Y

2

Y1 (1)×Y2 (1)

Figure 2. Excision of overlapping products when n = 2. The proof is clear from the picture. We apply the Q lemma to a pair of geodesic Q product-simplices in flat-hyperbolic space. Let ∆ = i∈I ∆i and ∆′ = i∈I ∆′i ni where ∆i , ∆′i are geodesic simplices in Eni or H for each i ∈ I. By the lemma, ∆ ∪ ∆′ can be decomposed as a disjoint union of products of ∆i ∩ ∆′i ,

∆i \(∆i ∩ ∆′i ) ,

or

∆′i \(∆i ∩ ∆′i ) ,


27

for i ∈ I. In each case, the intersection of two geodesic simplices (or its complement) is a geodesic polytope. Every such polytope can be decomposed as a disjoint union of geodesic simplices by subdividing and triangulating. It follows that we can triangulate any overlapping union ∆ ∪ ∆′ with geodesic product-simplices. Corollary 4.9. Any finite union of product-simplices can be obtained by gluing finitely many geodesic product-simplices along pairs of common facets. We can now prove the main result of this section, following [65]. Proposition 4.10. Every product-hyperbolic manifold M n of finite volume admits a finite tiling by products of geodesic simplices. S Proof. By theorem 4.1, there is a finite decomposition M n = ℓ M(ℓ) , where M(ℓ) is a cone over a compact flat-hyperbolic manifold with boundary Fℓ . We first tile each piece M(ℓ) by tiling Fℓ and then taking cones at infinity over this tiling. We then obtain a tiling of M n on taking the union of all the geodesic product-simplices involved and excising any overlaps using the corollary above. ByS§4.1.1, there is a fundamental set F ⊂ Xn for M n , with a decomposition diffeomorphic Q to Rk>0 × Dℓ , and Dℓ is a compact domain F = ℓ Fℓ , where Fℓ is Q ni −1 × j∈I\S Hnj . Let π : Xn → M n denote the in flat-hyperbolic space i∈S E covering map. We cover Dℓ with geodesic product-simplices as follows. Since Dℓ is compact, we firstQ choose compact sets Ki ⊂ Eni −1 , Kj ⊂ Hnj , where Q i ∈ S, j ∈ I\S, such that Dℓ ⊂ i∈S Ki × j∈I\S Kj . We denote the restriction of the covering map Dℓ → Fℓ by π also. Each set Ki can be triangulated by finitely (i) many geodesic simplices ∆ai which are sufficiently small such that any product Q Q (j) (i) ∆a = i∈S ∆ai × j∈I\S ∆aj , where a = (ai )i∈I , is mapped isometrically onto π(∆a ). Let a 6= b be indices such that the pair of simplices π(∆a ), π(∆b ) have nonempty overlap. It follows that there is a γ ∈ Γ such that γ∆a ∩ ∆b 6= ∅. Applying the previous corollary to the union γ∆a ∪ ∆b , we can replace ∆a ∪ ∆b with a union of product-simplices whose interiors are disjoint after projection down to F ℓ . We can repeatedly excise the overlap between simplices to obtain the required tiling of Dℓ with product-simplices. To show that this process terminates after finitely many steps, let d(x) ∈ N denote the multiplicity of the tiling at each point x ∈ Dℓ . Since F is a fundamental set, there is an integer N such that d(x) ≤ N for all x ∈ Dℓ . Every time an excision is applied, the local multiplicities d(x) for all x in some open subset of Dℓ decrease by 1. Since Dℓ is compact, this process terminates when d(x) = 1 almost everywhere. This gives a tiling of Dℓ . We obtain a tiling of the end Fℓ by taking the cone at infinity, i.e., we take every product of flat-hyperbolic geodesic simplices that occurs in the tiling of Dℓ : Y Y Y Y Hnj , Eni −1 × ∆(j) ∈ ∆(i) × i

j

i

j

and replace it with simplices obtained by adding the points at infinity: Y Y Y Y nj H . Eni −1 × R ∪ {∞} × ∆(j) ∈ (∆(i) , ∞) × i

i

j

n

j

We lift this to a tiling of Fℓ ⊂ X in product-hyperbolic space by identifying ni Eni −1 × R with a suitable horoball in H . This gives a covering of each cusp sector Fℓ by finitely many products of geodesic product-hyperbolic simplices such that its image under π gives a tiling of M(ℓ) . However, this does not give a tiling of

28

FRANCIS C.S. BROWN

Fℓ , since the tiling may extend into other cusp sectors of F . We can remove the overlaps by applying the same excision argument as above. Since any overlaps are necessarily compact, we obtain a genuine tiling after finitely many steps. The proof of the theorem gives a tiling and not a triangulation, since the subdivisions used in the excision lemma do not necessarily extend to neighbouring simplices. Note also that the geodesic simplices which occur have at most one vertex at infinity, a fact which will be used later. It does not matter if degenerate simplices occur in the tiling. The proof also works for any manifold modelled on products of spaces of constant curvature. Figure √ 3 below shows a product-triangulation of the Euclidean equivariant lattice in Z[ 2] ⊂ R2 .

111111 000000 000000 111111 000000 111111 000000 111111 0000001111111 111111 0000000 00000 11111 0000000 1111111 00000 11111 0000000 1111111 00000 11111 0000000 1111111 00000 11111 0000000 1111111 00000 11111 0000000 1111111 00000000 11111111 00000 11111 0000000 1111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

b

c

a

e e

c

d

d

f

f

a b

Figure√3. The √ equivariant Euclidean lattice whose vertices are {(a + b 2, a √ − b 2) : a, b ∈ Z}. the left is √ On √ √ a product tiling with tiles [0, 2] × [0, 1] and [ 2 − 1, 2] × [− 2, 0], and on the right is the corresponding tiling of the plane. Also shown are the six false faces on each tile, with their gluing relations. Remark 4.11. Given a finite virtual tiling of M n , one can obtain a genuine tiling by subdividing and excising any overlaps using corollary 4.9. 4.5. Deformation of product-tilings. We will need to move the vertices which define a product-tiling so that they lie in a fixed number field. One cannot move each vertex individually since this does not preserve the product structure of each tile, nor does it preserve the gluing relations between adjacent tiles. In order to deform a tiling, one must move several vertices which lie along a geodesic hyperplane simultaneously. Suppose z ∈ Hni is a point in the ith component of Xn . If z ′ is sufficiently near to z, we can replace every product of simplices of the form ∆1 × . . . × ∆i−1 × ∆(z, x1 , . . . , xni ) × ∆i+1 × . . . × ∆N ⊂

N Y

i=1

Hni ,


29

which occurs in the tiling of M n , with a similar product obtained by replacing z with z ′ . Here, ∆(z, x1 , . . . , xni ) is the geodesic simplex with vertices at the point z and another ni points x1 , . . . , xni ∈ Hni , and ∆1 , . . . , ∆N are geodesic simplices in Hni . One can do the same thing for geodesic simplices with coordinates at infinity, provided that we only move the coordinates of finite vertices. Equivalently, we can move a maximal geodesic polytope P obtained by gluing a finite number of adjacent faces Fi together. Such a polytope P can be deformed transversally without affecting the combinatorics of the tiling (see below). If P passes through another polytope, then this makes a crease, and we obtain a virtual tiling (fig. 1).

1 0 0 1

F1 1 0 0 1

1 0 0 1

F2

11 00 00 11

P

Let S = (k1 , . . . , kN ) denote an ordered set of fields as in §4.2. Corollary 4.12. Suppose that M n is defined over S. Then M n admits a tiling with product simplices whose vertices are S-rational. Proof. The set of S-rational points is dense in M n . We know by lemma 4.9 that coordinates at infinity in the cusps of M n are S-rational. By deforming the triangulation, we can ensure that all remaining vertices of each geodesic simplex ∆i ⊂ Hni lie in the set of ki -rational points of Hni , for each i ∈ I. Since the displacements can be made arbitrarily small, there is no creasing and we have a genuine tiling. 4.5.1. Equivariance. Now suppose that the product-hyperbolic manifold M n is equivariant. By the previous corollary, there exists a product-tiling of M n which is defined over S, where S = (σ1 k, . . . , σN k). Let T denote the set of hyperbolic product-simplices in this tiling. The symmetry group SN acts on the productsimplices, and preserves the subdivisions of a face F into subfaces Fi : F =

m X i=1

Fi

implies that

σ

F =

m X

σ

Fi ,

i=1

for all σ ∈ SN . Note that the simplices σFi may be oriented negatively, so the righthand sum is a virtual tiling of σF . Let σT denote the set of images of the elements of T under σ ∈ SN . We can glue the simplices in σT back together according to the same gluing pattern to form a virtual tiling of σM n . Since M n was assumed to be equivariant, this gives a new tiling of M n , which is also defined over S. Lemma 4.13. Let M n denote an equivariant product-hyperbolic manifold. If T is a product-tiling of M n defined over S, then σ T is also a virtual product-tiling of M n defined over S, for all σ ∈ SN .

30

FRANCIS C.S. BROWN

4.6. Scissors congruence groups and the generalised Dehn invariant. This section and the next are not required to prove theorems 1.1 and 1.2. The scissors congruence group of a space of constant curvature is the free abelian group generated by geodesic simplices modulo cutting and gluing. Let V n denote En , Sn or Hn , and let G denote the Q-points of the corresponding group of isometries (which are not necessarily assumed to be orientation-preserving). Let α denote an orientation of V n , and let −α denote the opposite orientation. Definition 4.14. The scissors congruence group P(V n ) is the free abelian group on the symbols {∆(x0 , . . . , xn ), α}, where x0 , . . . , xn ∈ VQn , modulo the relations: (1) {∆(g x0 , . . . , g xn ), g α} = {∆(x0 , . . . , xn ), α} for all isometries g ∈ G. (2) For any n + 2 points x0 , . . . , xn+1 ∈ Vn , X (−1)i {∆(x0 , . . . , x bi , . . . , xn+1 ), α} = 0. 0≤i≤n+1

(3) {∆(x0 , . . . , xn ), α} = 0 if x0 , . . . , xn lie on a geodesic hyperplane. (4) If Sn+1 denotes the symmetric group acting by permutation of the coordinates, then {∆(xπ(0) , . . . , xπ(n) ), α} = −ε(π){∆(x0 , . . . , xn ), −α} for any π ∈ Sn+1 with signature ε(π).

The natural action of Gal(Q/Q) on VQ gives rise to a well-defined action on P(V n ).

Now consider the case of hyperbolic space Vn = Hn . Following [53], we den fine a scissors congruence group P(H ) to be the free abelian group on generators n {∆(x0 , . . . , xn ), α} where we now take x0 , . . . , xn+1 to lie in HQ , modulo the relations (1)-(4) above. There is an inclusion: n

P(Hn ) → P(H ).

(4.6)

Theorem 4.15. (Sah) The map (4.6) is an isomorphism of abelian groups. The theorem says that a hyperbolic simplex with some vertices at infinity is stably scissors congruent to a sum of finite hyperbolic simplices, but it does not mean that there is an actual decomposition into finite simplices. Let ωn denote the volume form on Hn . The signed volume of {∆, α} is the integral of ωn over the oriented simplex ∆. This may be positive or negative, and is always finite. This defines a homomorphism n

vol : P(H ) → R which is compatible with (4.6). We need to consider the scissors congruence groups Q of products of hyperbolic simplices. If Xn = i∈I Hni , we can define:

(4.7)

(4.8)

P(Xn ) n

P(X )

= P(Hn1 ) ⊗ . . . ⊗ P(HnN ) , n1

= P(H ) ⊗ . . . ⊗ P(H

nN

).

Theoreom 4.15 implies that the natural inclusion is an isomorphism: ∼

(4.9) n

n

P(Xn ) −→ P(X ) .

The volume map vol : P(X ) → RQ is defined on generators {∆1 , α1 } ⊗ . . . ⊗ {∆, αn } n to be the product of the volumes i=1 vol({∆, αi }). This is compatible with (4.9).


31

4.6.1. The Dehn invariant for products of hyperbolic spaces. The Dehn invariant is defined for all spaces of constant curvature, but here we consider the hyperbolic case only. It is a homomorphism: Dn : P(Hn ) →

n−2 M i=1

P(Hi ) ⊗ P(Sn−i−1 ) ,

n

defined on generators {∆, α} of P(H ) as follows [24]. An i-dimensional face F of ∆ defines a geodesic plane in Hn which is isometric to Hi . The face F defines a geodesic simplex ∆F inside it. After choosing an orientation αF of this plane, we obtain an element {∆F , αF } ∈ P(Hi ). Next, we define an element ∆F ⊥ ∈ P(Sn−i−1 ) by choosing a point x in the interior of ∆F and considering the geodesic (n − i)-plane orthogonal to F passing through x. Since a sphere in Hn is a isometric to Sn−1 , the intersection of ∆ with a small sphere centered at x defines a spherical simplex ∆F ⊥ (with the ordering on the vertices induced by that of ∆). This simplex is given the orientation αF ⊥ which is compatible with the other two, i.e., αF ⊗ αF ⊥ = α. The Dehn invariant is defined by: X Dn ({∆, α}) = {∆F , αF } ⊗ {∆F ⊥ , αF ⊥ } .

where the sum is over all faces F of ∆ of codimension ≥ 2. In odd-dimensions n = 2k − 1, there is an extended Dehn invariant for the infinite scissors-congruence n group P(H ) whose definition is due to Thurston: n

Dn : P(H ) →

n−2 M i=1

i

P(H ) ⊗ P(Sn−i−1 ) .

It is defined in an identical manner to the ordinary Dehn invariant Dn in all components corresponding to edges of dimension i ≥ 2. The difference is that the one-dimensional edges (which may have infinite length) have to be regularised (see [53], Appendix 2 for details). The maps Dn and Dn are compatible with (4.6). Q ni Definition 4.16. Let Xn = N i=1 H . The generalised Dehn invariant is the map: Dn : P(Xn ) →

N nM k −2 M k=1 i=1

P(Hn1 ×. . .×Hnk−1 ×Hi ×Hnk+1 ×. . .×HnN )⊗P(Snk −i−1 ),

P where Dn is the tensor product k∈I id ⊗ . . . ⊗ Dnk ⊗ . . . ⊗ id. If the ni are all odd, n let Dn denote the corresponding generalised Dehn invariant on the space P(X ). 4.7. The scissors-congruence class of a product-hyperbolic manifold. Let M n = Xn /Γ be a product-hyperbolic manifold of finite volume defined over fields S = (k1 , . . . , kN ). By the tiling theorem, there is a finite tiling T of M n with product-hyperbolic simplices whose vertices lie over S. By corollary 4.12, the scissors-congruence class of T defines an element n

ξ M n ∈ P(X ) . It is well-defined because a different tiling defines a class which is scissors-congruent n to ξ M n . Now, using the Sah isomorphism P(Xn ) → P(X ) , we can pull ξ M n back to define a class ξM n ∈ P(Xn ), the scissors-congruence class of M n .

32

FRANCIS C.S. BROWN

Theorem 4.17. Let M n = Xn /Γ be a product-hyperbolic manifold of finite volume defined over S. Assume that M n has cusps in hyperbolic components of odd dimensions only. Then the scissors-congruence class ξM n ∈ P(Xn ) is invariant under the natural action of Gal(k 1 /k1 ) × . . . × Gal(k N /kN ) and satisfies: Dn (ξM n ) = 0 ,

and

vol(ξM n ) = vol(M n ) .

Now let us suppose that M n is equivariant with respect to a set of embeddings Σ = {σ1 , . . . , σN } of k. Choose any extension σj σi−1 : Q → Q of the isomorphisms σj σi−1 : ki → kj . This induces a twisted action of SN on P(X) as follows: −1 −1 π({∆1 , α1 }⊗. . .⊗{∆n , αn }) = ε(π)({σ1 σπ(1) ∆π(1) , α1 }⊗. . .⊗{σn σπ(n) ∆π(n) , αn }) ,

for all π ∈ SN . Then the action on ξMn is independent of the choices made, and ξM n = ξM n ∈ P(Xn )

π

for all

π ∈ SN .

In particular, Dn (πξM n ) = 0 and vol(π ξM n ) = vol(M n ) for all π ∈ SN . Proof. By construction, ξM n is well-defined, and vol(ξM n ) = vol(M n ). It suffices to show that the generalised Dehn invariant vanishes on ξM n , which follows from the fact that M n has no boundary. We show that Dn (ξ M n ) = 0. Consider any product Q (i) of simplices ∆j = i∈I ∆j which occurs in a given tiling T of M n constructed (k)

above, and let Fk be a facet in one component ∆0 . Then we must check that (1)

(k−1)

F = ∆0 × . . . × ∆0

(k+1)

× Fk × ∆0

(n)

× . . . × ∆0

does not occur in the image of the generalised Dehn invariant. Let SF denote the set of product-simplices in T which meet F . We can assume, by subdividing locally, that F is a product of faces in each element of SF (i.e., locally we have a triangulation, and not just a tiling). Then in the sum X X X (1) (n) (k) Dn (∆j ) = ∆j ⊗ . . . ⊗ Dnk (∆j ) ⊗ . . . ⊗ ∆j , j∈SF

j∈SF k∈I

F can only occur in the k-component id ⊗ .. ⊗ Dnk ⊗ .. ⊗ id of Dn . Its coefficient is X (1) X (n) ∆0 ⊗ . . . ⊗ Fk ⊗ θj (Fk ) ⊗ . . . ⊗ ∆0 = F ⊗ θj (Fk ), j∈SF

j∈SF

(k)

where θj (Fk ) ∈ P(Sl ) is the angle subtended at the face Fk in ∆j , for some l. Since M n is a manifold and P the tiling is proper, the sum of these angles is 2π for each k ∈ I, i.e., we have j∈SF θj (Fk ) = 0 ∈ P(Sl ). It follows that all faces, including those with vertices at infinity, occur with coefficient 0, and hence Dn (ξ M n ) = 0. Since (4.9) is compatible with Dn , we have Dn (ξM n ) = 0. Now, we showed in corollary 4.12 that Mn admits a tiling T which is defined over the set of fields S. Therefore T is Gal(k 1 /k1 ) × . . . × Gal(k N /kN )-invariant, and so too must be its scissors-congruence class ξM n . In the case when M n is equivariant, σT is a virtual tiling of M n by lemma 4.13 for all σ ∈ SN . The scissors-congruence class of σT is σ ξM n , which therefore coincides with ξM n .


33

4.8. Poincar´ e’s formula for even-dimensional simplices. Let n be an even number, and let ∆ be a geodesic simplex in Hn . For each i-dimensional face F of ∆, choose an interior point x ∈ F . Define θ(F ) to be the n − 1-dimensional spherical polytope formed by the closure of the set of vectors with origin at x which are in the interior of ∆. This does not depend on the choice of point x. If we set σk = vol(Sk ), then Poincaré’s formula ([1], §2.4) states that σn X (−1)dim F vol(θF ) . (4.10) vol(∆) = 2σn−1 F

The formula remains valid for simplices with vertices at infinity on setting θF = 0 for any such F ∈ ∂Hn . When n = 2, it gives the formula for the area of a geodesic hyperbolic triangle with interior angles α, β, and γ: vol(∆) = α + β + γ − 3π + 2π.

4.8.1. Proof of theorem Now consider a product-hyperbolic manifold M n = Q 2n3.8. n n X /Γ where X = i H i consists of even-dimensional spaces only. The argument in the proof of theorem 4.17 shows that the sum over all angles θF in a tiling of M n around each face F is a multiple of σn−1 . It follows that (4.11)

vol(ξM n ) ∈ π

P

2ni

Q× .

34

FRANCIS C.S. BROWN

5. The motive of a product-hyperbolic manifold We show how a triangulated product-hyperbolic manifold defines an element in the category of mixed Tate motives over a number field. This builds on ideas of Goncharov [29], and [6]. 5.1. Mixed Tate motives and the Hodge regulator. We recall some standard properties of framed mixed Tate motives and their periods [22, 6, 30]. 5.1.1. Mixed Tate motives over a number field. Let k be a number field, and let MT(k) denote the category of mixed Tate motives over k. It is a rigid abelian tensor category, and it is the smallest abelian subcategory of the derived category of motives which is generated by the Tate objects Q(n), and closed under extensions. The relation with algebraic K-theory is the isomorphism: 0 if n ≤ 0 , 1 ∼ (5.1) Ext (Q(0), Q(n)) = K2n−1 (k) ⊗ Q , if n ≥ 1. One has K1 (k) = k × , and it is known by Borel’s theorem [10], that for n > 1, r1 + r2 , if n is odd ; (5.2) dimQ (K2n−1 (Q) ⊗ Q) = n± = r2 , if n is even , where r1 is the number of real places of k, and r2 is the number of complex places of k. Although we do not explicitly require (5.2) to prove theorem 1.1, it is used implicitly in the construction of MT(k) (the Beilinson-Soulé vanishing conjecture). Every mixed Tate motive M ∈ MT(k) has a canonical weight filtration, and we denote its graded pieces by grW −2n M , for n ∈ Z. Following [22], one sets ωn (M ) = Hom(Q(n), grW −2n M ) .

L Then ω = n ωn is a fiber functor to the category of finite-dimensional graded Qvector spaces. One defines the de Rham realisation of M to be MDR = ω(M ) ⊗Q k. Then, for every embedding σ : k → C into an algebraic closure C of k, let Mσ denote the corresponding Betti realisation. There is a comparison isomorphism (5.3)

∼

compσ,DR : MDR ⊗k,σ C −→ Mσ ⊗Q C ,

which is functorial with respect to σ. The data consisting of (MDR , Mσ , compσ,DR ) for each σ : k ֒→ C defines a Q-mixed Hodge-Tate structure we denote Hσ . Its underlying complex vector space is MDR ⊗k,σ C, its weight and Hodge filtrations are induced from the grading on MDR , and its Q-structure comes from pulling back the Q-structure on MσQvia the comparison map (5.3). The Hodge realisation functor is the map M 7→ σ:k→C Hσ to the category of Q-mixed Hodge-Tate structures. It is proved in [22] that the Hodge realisation functor is fully faithful, although we shall not require this fact. Since ω is a fiber functor, the category MT(k) is Tannakian with pro-algebraic structure group GMT(k) . The grading defines a split exact sequence 0 → Gm → GMT(k) → UMT(k) → 0 , where the group UMT(k) is pro-unipotent.


35

5.1.2. Framed objects in MT(k). We recall the notion of framed mixed Tate motives from [6, 31]. Let M ∈ MT(k), and let n ≥ 0. Definition 5.1. An n-framing of M consists of non-zero morphisms: v0 fn

∈

∈

ω0 (M ) = Hom(Q(0), grW 0 M) , ωn (M )∨ = Hom(grW −2n M, Q(n)) .

Consider two n-framed objects (M, v0 , fn ), and (M, v0′ , fn′ ). They are said to be equivalent (written (M, v0 , fn ) ∼ (M ′ , v0′ , fn′ )), if there is a morphism φ : M → M′

such that φ(v0 ) = v0′ and fn′ (φ) = fn . This generates an equivalence relation on the set of n-framed objects. The equivalence classe of (M, v0 , fn ) is written [M, v0 , fn ]. Let An (k) denote the set of equivalence classes of all n-framed objects in MT(k). One easily shows that An (k) is a Q-vector space with respect to the addition rule: (5.4)

[M, v0 , fn ] + [M ′ , v0′ , fn′ ] = [M ⊕ M ′ , v0 ⊕ vn′ , fn + fn′ ] ,

and scalar multiplication α.[M, v0 , fn ] = [M, α.v0 , fn ] = [M, v0 , α.fn ] for all α ∈ Q× . One verifies that the zero element is given by the class of Q(0) ⊕ Q(n) with trivial framings, and that A0 (k) ∼ = Q. Consider the graded Q-vector space: M (5.5) A (k) = An (k) . n≥0

It has the structure of a graded ring via the commutative multiplication map

′ ′ [M, v0 , fn ] × [M ′ , v0′ , fm ] = [M ⊗ M ′ , v0 ⊗ v0′ , fn fm ], L and has a coproduct ∆ = n≥0 ∆n , where M (5.7) ∆n : An (k) −→ Ar (k) ⊗Q An−r (k)

(5.6)

0≤r≤n

is defined on an element [M, v0 , fn ] ∈ An (k) as follows. Let {ei }1≤i≤N denote any ∨ basis in ωk (M ), and let {e∨ i }1≤i≤N denote the dual basis in ωk (M ) . Define (5.8)

∆k,n−k [M, v0 , fn ] =

N X i=1

[M, v0 , e∨ i ] ⊗ [M, ei , fn ](−k) ,

where [M, ei , fn ](−k) is the Tate-twisted object M (−k) with corresponding framL ings. One verifies that ∆n = 0≤k≤n ∆k,n−k does indeed give a coproduct. One defines a counit ε : A (k) → Q by projection onto the graded component A0 (k), and a unit map i : Q ∼ = A0 (k) ֒→ A (k). One proves that the above operations are well-defined and that A (k) is a graded commutative Hopf algebra (its antipode is uniquely determined from the above data). The main result is the following [30]. Proposition 5.2. A (k) is isomorphic to the dual of UMT(k) . In other words, one can identify A (k) with the Hopf algebra of functions on UMT(k) . The duality is given by the following map:

(5.9)

UMT(k) × A (k)

(φ, [M, v0 , fn ])

−→ Q 7→

hφ(v0 ), fn i .

The right-hand term is a rational number, because φ ∈ UMT(k) defines an endomorphism of the Q-vector space ω(M ). The proposition states that (5.9) is a

36

FRANCIS C.S. BROWN

e on A (k) by the formula perfect pairing. Now define the reduced coproduct ∆ e ∆(X) = ∆(X) − 1 ⊗ X − X ⊗ 1. Its kernel is the set of primitive elements in A (k).

Corollary 5.3. There is an isomorphism: M e n : An (k) −→ Ar (k) ⊗Q An−r (k) . (5.10) Ext1MT(k) (Q(0), Q(n)) ∼ = ker ∆ 0≤r≤n

This follows immediately from the previous proposition, the finite-dimensionality of Ext1 (equations (5.1) and (5.2)), and proposition A.15 in [22]. The corollary gives a strategy for constructing elements in rational algebraic Ktheory, which goes back to Beilinson and Deligne. The essential idea is to write down a sum of framed mixed Tate motives, using the cohomology of simple algebraic varieties, in such a way that the reduced coproduct vanishes. 5.1.3. Real periods. To each framed mixed Tate motive [M, v0 , fn ] ∈ An (k), one can associate a set of numbers Rσ indexed by the set of complex embeddings of k, in the following way [29]. Consider the map M ∼ Pσ : ω(M ) = ωn (M ) −→ Mσ ⊗k,σ C , n

obtained by composing compσ,DR with the inclusion of ω(M ) into MDR ⊗k,σ C, and let Pσ∗ = (Pσ∨ )−1 denote the inverse dual map from to ω(M )∨ to (Mσ ⊗k,σ C)∨ . The algebraic closure C of k has a natural conjugation, and enables us to define the real and imaginary parts of any element in C, which lie in R. Definition 5.4. The real period corresponding to σ : k ֒→ C is the map: (5.11)

Rσ : An (k) −→ R Rσ ([M, v0 , fn ]) = hIm (2iπ)−n Pσ v0 , Re (Pσ∗ fn )i .

One proves that the real period is well-defined, and is a homomorphism for the multiplication on A (k). A detailed treatment is given in [29], §4. Here we have normalized the real period so that it gives a real number for all n. It differs by a factor of (2iπ)n from the version given in [29]. Definition 5.4 has the advantage of being basis-free. For calculations, however, one can choose Q-bases of ω(M ) and Mσ which are compatible with the weight filtrations, and write Pσ as a period matrix. We can assume without loss of generality n that each graded piece Pσ : ωn (M ) ∼ = grW −2n (Mσ ⊗k,σ C) is equal to (2πi) times the identity. The matrix Pσ is well-defined up to left multiplication by a unipotent lower-triangular matrix with rational entries. In this way, one checks that (5.12)

Rσ : Ext1MT(k) (Q(0), Q(n)) −→ R ,

is given on the level of period matrices by the map: α (2iπ)n 0 Pσ = , where α ∈ C/(2iπ)n . 7→ Im α 1 (2πi)n

Concretely, a dual period matrix Pσ∨ is obtained by integrating a basis of the de Rham cohomology of an algebraic variety against a basis of its homology. It is clear from the definition of the real period that Rσ τ = Rστ , for all τ ∈ G.


37

5.1.4. Determinants and the Hodge regulator. Let k be a finite extension of Q, and let Σ = {σ1 , . . . , σN } denote N distinct embeddings of k into an algebraic closure C. If we write ki = σi (k), there is a natural isomorphism: (5.13)

ρ=

N O i=1

σi : An (k) ⊗ . . . ⊗ An (k)−→An (k1 ) ⊗ . . . ⊗ An (kN ) .

Let An (k)⊗N denote the left-hand side of (5.13), viewed as a SN -module for the natural action of the symmetric group which permutes the N factors. Let An (k)⊗Σ denote the right-hand side of (5.13), with the induced SN -action: −1 −1 (Miπ(r) ) (Miπ(1) ) ⊗ . . . ⊗ σr σπ(r) π(Mi1 ⊗ . . . ⊗ Mir ) = σ1 σπ(1)

for π ∈ SN .

Now each embedding σ : k → ki ⊂ C defines a real period map Rσi on An (k). We define the total real period to be the product: RΣ = Rσ1 × . . . × RσN : An (k1 ) ⊗ . . . ⊗ An (kN ) −→ R .

Let ε denote the alternating representation of SN . Then ρ induces an isomorphism: VN ∼ An (k) ∼ = hAn (k)⊗N , εiS −→ hAn (k)⊗Σ , εiS . N

N

Definition 5.5. We will denote the right-hand side by VN VΣ An (k) . An (k) = ρ

If, throughout, we replace An (k) with the subspace Ext1MT(k) (Q(0), Q(n)) annihilated by the reduced coproduct, we obtain an isomorphism VN ∼ VΣ Ext1MT(k) (Q(0), Q(n)) . (5.14) ρ: Ext1MT(k) (Q(0), Q(n)) −→

Definition 5.6. We obtain a regulator as follows. Suppose that k has r1 distinct real embeddings σ1 , . . . , σr1 and 2r2 distinct complex embeddings given by σr1 +1 , . . . , σr1 +r2 and their complex conjugates. Then let {σ1 , . . . , σr1 , . . . , σr1 +r2 } , if n is odd , Σ= if n is even . {σ1 , . . . , σr1 } , Now we know from (5.1) and (5.2) that dimQ Ext1MT(k) (Q(0), Q(n)) = |Σ| = 1 n± in both cases, so the nth ± exterior power of ExtMT(k) (Q(0), Q(n)) is just its determinant, which is a Q-vector space of dimension 1. In this case we can also write VΣ Ext1MT(k) (Q(0), Q(n)) . detΣ Ext1MT(k) (Q(0), Q(n)) for Consider the Hodge regulator map: (5.15)

∼

RΣ ◦ ρ : det Ext1MT(k) (Q(0), Q(n)) −→ detΣ Ext1MT(k) (Q(0), Q(n))−→R .

Its image is a one-dimensional Q-lattice in R. We define the Hodge regulator to be its covolume. This defines a number R(k) ∈ R/Q× .

The Hodge regulator rH : Ext1MT(k) (Q(0), Q(n)) → Rn± given in the introduction L (1.3) is precisely the map rH = σ∈Σ Rσ . Its image is a Q-lattice whose covolume is R(k) mod Q× .

Remark 5.7. By (5.1), we could compare the Hodge regulator map with Borel’s regulator by identifying det (K2n−1 (k) ⊗ Q) ∼ = det (Ext1MT(k) (Q(0), Q(n))). It is generally expected that they should coincide up to some explicit rational number, although we do not know of a direct proof.

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FRANCIS C.S. BROWN

5.2. Construction of the motive of a product-hyperbolic manifold. We define the framed motive of a triangulated product-hyperbolic manifold by summing the motives of each geodesic simplex in the triangulation. To do this, it suffices to compute the framed motive of a simplex with at most one vertex at infinity. 5.2.1. The motive of a finite geodesic simplex. First of all, consider the case of an oriented hyperbolic geodesic simplex ∆ ⊂ Hm with no vertices at infinity. Goncharov’s idea is that it defines a mixed Tate motive, in the following way. In the Klein model for Hm , ∆ corresponds to a Euclidean simplex inside the unit sphere, which represents ∂Hm . After complexifying and compactifying, the unit sphere inside Rm becomes a smooth quadric Q inside PSm , and the simplex ∆ is given by a set of hyperplanes {L0 , . . . , Lm }.T Let L = i Li , and note that L ∪ Q is a normal crossing divisor. Write LI = i∈I Li for any subset I ⊂ {1, . . . , N }, and QI = Q ∩ LI . Consider the complex: a a a LI \QI . LI \QI ⇇ . . . ⇇ LI \QI ⇇ (5.16) Pm \Q ← |I|=m

|I|=2

|I|=1

If Q, L are defined over Q, (5.16) defines a mixed Tate motive (5.17)

h(∆) = H m (Pm \Q, L\(L ∩ Q)) .

For any subset J ⊂ {1, . . . , N }, we obtain a subcomplex by intersecting with LJ : a a LI∪J \QI∪J , LI∪J \QI∪J ⇇ . . . ⇇ (5.18) LJ \QJ ← |I∪J|=2n−1

|I∪J|=2r+1

The complex (5.18) corresponds to the face ∆ ∩ LJ inside LJ \QJ . The inclusion of (5.18) into (5.16) defines a morphism h(∆ ∩ LJ ) → h(∆). Suppose that m = 2n − 1 is odd. There is a canonical class (5.19)

∼

ωQ : Q(−n) −→ H 2n−1 (P2n−1 \Q) ,

whose residue along Q is the difference between the two canonical classes in H 2n−2 (Q), and whose sign is determined by an orientation on Q. Its de Rham cohomology class can be written explicitly (see (5.38) below) and corresponds to the volume form on H2n−1 in the Klein model. One verifies that (5.20)

∼ 2n−1 (P2n−1 \Q) ∼ grW = Q(−n) , 2n h(∆) = H

∼ W 2n−1 (P2n−1 , L) ∼ and grW = Q(0). By computing the spectral sequence 0 h(∆) = gr0 H of (5.16), one proves that, in the remaining weights, M Q(r − n) for 1 ≤ r ≤ n − 1 . h(∆) = (5.21) grW 2(n−r) |I|=2r

A basis for grW 2(n−r) h(∆) is given by the images of the classes (5.20) of all odddimensional faces. Thus, for each face F of dimension 2i − 1, the inclusion F ֒→ ∆ induces a map iF : h(F ) → h(∆) and we set

eF = iF ◦ ωQ∩F : Q(−i) −→ grW 2i h(∆) . The classes eF give a canonical basis for ω−i h(∆) . The details of these calculations are given in [29], and are repeated in the proof of proposition 5.8 below in a more complicated case (see also §5.3 for an example).

(5.22)


39

5.2.2. The motive of a simplex with a vertex at infinity. Now consider the case of an oriented hyperbolic simplex with a single vertex x on the absolute. As above, the simplex ∆ defines a configuration of hyperplanes L0 , . . . , Lm inside a smooth quadric Q. This time, the divisor Q ∪ L is not normal crossing, so we must blow-up the point x at infinity. Let us number the hyperplanes so that L1 , . . . , Lm intersect e m denote the at x, and L0 is the hyperplane which does not pass through x. Let P m e −1 denote the exceptional divisor, and let Q, e L e i denote the blow-up of P at x, let L strict transforms of Q, Li respectively. The complex (5.16) is now replaced with: a a a e m \Q e I \Q eI , e I \Q eI ⇇ . . . ⇇ e I \Q eI ⇇ e← L L L (5.23) P |I|=m

|I|=2

|I|=1

where I ⊂ {−1, 0, 1, . . . , m}. When (Q, L) are defined over Q, this defines a mixed Tate motive we will also denote: e m \Q, e L\( e L e ∩ Q)) e . (5.24) h(∆) = H m (P It turns out that this motive has an identical structure to the motive of a finite simplex h(∆) defined by (5.17), except in graded weight 2. This corresponds to the fact that the one-dimensional faces of ∆ passing through x have infinite length. We x

e −1 L

e2 L

e1 L

e0 L

Q Figure 4. A hyperbolic 3-simplex ∆ with one vertex at infinity x in the Klein model. After blowing up the point x, the exceptional e −1 meets the faces of ∆ e in a Euclidean triangle. divisor L

now describe elements in grW • h(∆) which will prove below to be a basis. As in the case of a finite simplex, there is a canonical class ∼ ∼ H 2n−1 (P e 2n−1 \Q) e . (5.25) ω e : Q(−n) −→ H 2n−1 (P2n−1 \Q) = Q

Similarly, for each face F of odd dimension 2i − 1 ≥ 3, the inclusion of the face F ֒→ ∆ defines a map iFe : h(F ) → h(∆) and gives rise to a map: (5.26)

W eFe : iFe ◦ ωQ∩ e F e : Q(−i) −→ gr2i h(∆) .

The situation is more subtle for one-dimensional faces. Let F be a one-dimensional face which does not pass through x, and so F ⊂ L0 . Then (F ∩ Q) ⊂ F is a quadric ∼ 1 1 ∼ in P1 , and grW 2 h(F ) = H (P \{2 points}) = Q(−1). Thus we can define a map eFe : Q(−1) → grW h(∆) in an identical manner to (5.26). This cannot work for 2 e in one-dimensional faces passing through x, because their strict transforms meet Q 1 1 a single point, and the corresponding classes vanish since H (P \{1 point}) = 0.

40

FRANCIS C.S. BROWN

What happens instead is the following. Let G be a 2-dimensional face of ∆ which meets x. Its strict transform corresponds to the complement of the blow-up of a e 2 \Q e G . There is a canonical class smooth quadric in the projective plane P (5.27)

∼ e 2 \Q eG) ∼ ηQeG : Q(−1) −→ H 2 (P = grW 2 h(G) ,

which we define below. As previously, the inclusion of the face G ֒→ ∆ induces a map iGe : h(G) → h(∆), and we define: (5.28)

αGe = iGe ◦ ηQeG : Q(−1) −→ grW 2 h(∆) .

Now consider the set of all faces L1 ,. . . , Lm which pass through the point x. Their e1, . . . , L e m intersect L e −1 in an oriented m − 1 simplex ∆∞ . This strict transforms L can be thought of as the intersection of ∆ with a horosphere around the point x. It defines a simplicial complex M M M Q, Q → ... → Q→ (5.29) Q→ |I|=1

|I|=m−1

|I|=2

where each indexing set I ⊂ {1, . . . , m}. The cohomology of this complex is that of an m − 1 sphere. By augmenting the complex, one obtains a Q-vector space M Q −→ Q) , (5.30) Vx = ker( |I|=m−1

spanned by all linear combinations of 1-dimensional faces of ∆ which meet x such that the sum of coefficients is 0. The face maps of ∆∞ give a section of (5.29) M M Q Q −→ |I|=m−1

|I|=m−2

which is injective on the subspace Vx . In this way, every one-dimensional edge of ∆ meeting x gives rise to a signed linear combination of 2-dimensional faces of ∆ meeting x. Finally, for each one-dimensional face F of ∆ meeting x, we define (5.31)

eFe : Q(−1) −→ grW 2 h(∆) .

to be the corresponding linear combination of maps αGe defined by (5.28). Clearly, X (5.32) eFe = 0 . F

where the sum is over all one-dimensional faces F of ∆ meeting x. As we shall see in the following section, this corresponds to the fact that there is a relation between the internal angles of the Euclidean simplex at infinity ∆∞ .

∼ 2n−1 (P2n−1 \Q) ∼ Proposition 5.8. If m = 2n − 1 is odd, grW = Q(−n), 2n h(∆) = H W 2n−1 e 2n−1 e ∼ W ∼ (P , L) = Q(0). In weights ≥ 4, and gr0 h(∆) = gr0 H M (5.33) Q(r − n) for 1 ≤ r ≤ n − 2 . grW 2(n−r) h(∆) = |I|=2r

A canonical basis is given by the classes eFe for all odd-dimensional faces F of ∆. In graded weight 2, we have (5.34) grW h(∆) ∼ = Q(−1)m(m+1)/2−1 , 2

which is spanned by the classes eFe for all m(m+1) one-dimensional faces F of ∆, 2 subject to the single relation (5.32) for the faces which meet the point at infinity x.


41

Proof. Recall that the cohomology of a smooth quadric Q of dimension ℓ vanishes in odd degrees, and satisfies H 2i (Q) = Q(−i) for 1 ≤ i ≤ ℓ, unless ℓ = 2k is even, in which case H 2k (Q) = Q(−k) ⊕ Q(−k) in middle degree [27]. When ℓ > 1, the strict e is just the blow-up of Q in the point x. From the Gysin sequence: transform Q e m ) −→ H i (P e m \Q) e e −→ H i−1 (Q)(1) e . . . −→ H i−2 (Q)(1) −→ H i (P −→ . . . , we deduce that (5.35)

 Q(−n) ,    Q(−1) , e m \Q) e = H i (P Q(0) ,    0,

if i = m is odd and equal to 2n − 1 ; if i = 2 ; if i = 0 ; otherwise .

e m \Q) e when m is odd, is the image of the canonical class The generator of H m (P e m \Q) e and is given by the difference of the two classes on the H m (Pm \Q) → H m (P quadric (see below). The complex (5.23) defines a spectral sequence converging to grW h(∆), with M e m \Q) e I \Q e I ) for q ≥ 1 , and E 0,q = H q (P e . H p (L (5.36) E1p,q = 1 |I|=q

e −1 \Q e −1 is isomorphic to affine space, and therefore H p (L e I \Q eI ) = 0 Observe that L for all I which contain −1, and all p ≥ 1. For all sets I containing the index e I \Q eI ∼ 0, L = LI \QI is the complement of a smooth quadric in projective space i and H (LI \QI ) vanishes when i > 0, unless |I| = 2r is even, and i = m − 2r. e I \Q e I ) is given by (5.35). The spectral sequence (5.36) For all other sets I, H p (L degenerates at E2 , and one deduces that for 1 ≤ r ≤ n − 2, M H 2(n−r)−1 (LI \QI ) . (5.37) grW 2(n−r) h(∆) = |I|=2r

To compute

grW 2 h(∆),

E12,k =

observe that M m Q(−1) ∼ = Q(−1)( k )

I⊂[m], |I|=k

for 0 ≤ k ≤ m − 2 .

where [m] denotes {1, . . . , m}. The complex 0 → E12,0 → E12,1 → . . . → E12,m−2 , is precisely the complex (5.29), and we deduce that its cokernel is E 2,m−2 ∼ = Q(−1)m−1 ∼ = Vx ⊗Q Q(−1) . 2

eI ∼ Finally, we consider the one-dimensional edges. If |I| = m − 1, then L = P1 . It e I in two points if 0 ∈ I, and in exactly one point if meets the blown-up quadric Q 1 e e I ) = 0, so we have 0∈ / I. In the latter case H (LI \Q M Q(−1) = Q(−1)m(m−1)/2 , E22,m−1 = E11,m−1 = 0∈I, |I|=m−1

since L0 ∩ L−1 = ∅. In total, we have ∼ 2,m−1 ⊕ E 2,m−2 ∼ grW = Q(−1)m(m+1)/2−1 . 2 h(∆) = E2 2

W 2n−1 e 2n−1 e ∼ We have grW (P \Q) = Q(−n), and also 2n h(∆) = gr2n−1 H M M Q(0) ∼ Q(0) → grW = Q(0) , 0 h(∆) = ker |I|=2n−1

|I|=2n−2

42

FRANCIS C.S. BROWN

since the hyperplanes L form a simplicial complex which has the cohomology of a sphere. This gives (5.33). In both cases, whether ∆ has a vertex at infinity or not, we have ∼ W 2n−1 (P2n−1 \Q) and grW h(∆) = grW H 2n−1 (P2n−1 , L) . grW 2n h(∆) = gr2n H 0 0

This enables us to define framings on h(∆) as follows. As we saw earlier, the class ωQ (respectively, ωQe ) gives a framing on H 2n−1 (P2n−1 \Q), and there is a ∼

canonical morphism grW 0 h(∆) → Q(0) given by the complex of faces L of ∆. In the Hodge realisation, we can write these P classes explicitly. Let x0 , . . . , x2n−1 denote coordinates on P2n−1 , and let q = ij aij xi xj be an explicit quadratic form defining the quadric Q. Then P2n c i ∧ . . . ∧ dx2n−1 p (−1)i xi dx0 ∧ . . . ∧ dx , (5.38) ωQ = ±in det q i=1 n q (x)

defines a class [ωQ ] which generates H 2n−1 (P2n−1 \Q). It does not depend on the ∨ choice of coordinates. The framing on grW 0 h(∆) is given by the relative homology 2n−1 class of the real simplex ∆, which defines a class [∆] ∈ grW , L) ∼ = 0 H2n−1 (P W ∨ gr0 h(∆) . Now suppose that ∆ has an orientation. If ∆ is positively oriented, we can, for example, normalise the signs of [∆] and [ωQ ] such that, Z vol(∆) = i1−n ωQ ∈ R ∆

is positive, otherwise, if ∆ is negatively oriented, we replace one of the framings with its negative so that vol(∆) < 0.

Definition 5.9. Let ∆ be an oriented hyperbolic geodesic simplex in H2n−1 , with at most one vertex at infinity. Suppose it is defined over a number field k. Let h(∆) be defined by (5.17) in the case where ∆ is finite, and by (5.24) in the case where ∆ has a vertex at infinity. We set p (5.39) mot(∆) = h(∆), [ωQ ], [∆] (n) ∈ An (k( | det q|)) .

Note that this framed motive can be defined completely motivically, i.e., with morphisms of complexes of varieties and without reference to the Hodge realisation. However, it will be more convenient for us to write the framings in terms of the Hodge realisation.

Remark 5.10. One can extend the definition to the case where the simplex ∆ is degenerate. In this case, mot(∆) is equivalent to the zero object. Likewise, in the even-dimensional case, the group H m (Pm \Q) vanishes by (5.35), and so there is no non-zero framing. Therefore, mot(∆) is non-zero only in the generic, odddimensional case. 5.2.3. Definition of the framed motive of a product-hyperbolic manifold. We first require the following subdivison lemma. m

Lemma 5.11. Let x0 , . . . , xm be distinct points in HQ in general position except that at most one may lie on the absolute, and let ∆(x0 , . . . , xm ) denote the geodesic , simplex whose vertices are given by the xi . Given any finite point y ∈ Hm Q (5.40)

mot(∆(x0 , . . . , xm )) =

m X i=0

mot(∆(x0 , . . . , xi−1 , y, xi , . . . , xm )) .


43

Proof. Suppose that x0 lies on the absolute, and assume that x0 , . . . , xm , y are in general position. Let Li , for i ∈ I, denote the set of m+2 hyperplanes spanned m by all m-tuples of points in {x0 , . . . , xm , y}, and let J ⊂ I denote the subset corresponding to m-tuples in {x0 , . . . , xm }. Let Q denote a smooth quadric in Pm corresponding to the absolute of Hm . After blowing-up the point x0 , and all intersections of hyperplanes which do not cross normally, we obtain a configuration e i of Li and the exceptional loci Lh of hyperplanes given by the strict transforms L m e denote the blow-up, and let Q e denote the strict indexed by some set H. Let P transform of Q. The complex a a e m \Q e← e i \(L e i ∩ Q) e ⇇ e ij \(L e ij ∩ Q) e ⇇ ... (5.41) P L L i∈I∪H

i,j∈I∪H

defines a mixed PmTate motive. There are unique framings on it such that it is equivalent to i=0 mot(∆(x0 , . . . , xi−1 , y, xi , . . . , xm )). Likewise, the complex a a e m \Q e← e i \(L e i ∩ Q) e ⇇ e ij \(L e ij ∩ Q) e ⇇ ... (5.42) P L L i∈J∪H

i,j∈J∪H

also defines a mixed Tate motive, which can be given unique framings making it equivalent to mot(∆(x0 , . . . , xm )). The inclusion of (5.42) into (5.41) is the required equivalence of framed motives, and corresponds to the fact that the relative homology class [∆(x0 , . . . , xm ))] is equal to the sum of the relative homology classes Pm i=0 [∆(x0 , . . . , xi−1 , y, xi , . . . , xm ))]. The proof is similar in the case where all points xi are finite, or when x0 , . . . , xm , y are not in general position, in which case, some of the framed motives in (5.40) will be zero. When m = 2n − 1, Goncharov proves a different, but equivalent, version of this lemma for finite simplices [29]: given m + 1 hyperplanes M0 , . . . , Mm in general position over Q, and letting ∆i denote the oriented geodesic simplex bounded by ci , . . . , Mm , one has Pm (−1)m mot(∆i ) = 0. the hyperplanes M0 , . . . , M i=0

Definition 5.12. Let M denote a finite-volume product-hyperbolic manifold modelled on Xn = H2n1 −1 × . . . × H2nN −1 and defined over the fields S = (k1 , . . . , kN ). By theorem 4.17, M admits a product-tiling M=

M X i=1

(i)

(i)

∆1 × . . . × ∆N , nj

(i)

where ∆j is a geodesic simplex in H which is defined over kj , and which has at most one vertex at infinity. The (framed) motive of M is defined to be the element (5.43)

mot(M ) =

M X i=1

(i)

(i)

mot(∆1 ) ⊗ . . . ⊗ mot(∆N ) ∈ An1 (k1 ) ⊗ . . . ⊗ AnN (kN ) .

It follows from lemma 5.11 that mot(M ) is well-defined, since one can construct a common subdivision of any two distinct product-tilings of M . Note that the definition requires that each component of Xn be of odd dimension, since the motive of a simplex in even-dimensional hyperbolic space is 0 by remark 5.10.

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FRANCIS C.S. BROWN

5.2.4. Vanishing of the reduced coproduct. We must next prove that mot(M ) is a product of extensions of Q(0) by Q(n). Let ∆ be a geodesic hyperbolic simplex defined over Q with at most one vertex at infinity. In §5.2.2 we constructed a basis of grW e , where F ranges over all faces of ∆ of 2i h(∆) in terms of the classes eF dimension 2i − 1. For every such face, we define (5.44)

mot(F ) = [h(∆), eFe , [∆]] ,

where, as usual, the framing coming from the homology class [∆] ∈ grW 0 h(∆) should be thought of as an appropriate morphism of complexes. If F is of dimension ≥ 3, or is of dimension one and does not meet a point at infinity x, then mot(F ) is equivalent to the motive of F , and the definition is compatible with (5.39). If F is one-dimensional and passes through x, then mot(F ) is a linear combination of motives coming from 2-dimensional faces of ∆, framed by the classes (5.28). With this definition, we can write the reduced coproduct of h(∆): X e ∆(mot(∆)) = mot(F ) ⊗ cF , F

where the sum is over all odd-dimensional strict subfaces F of ∆. We say that cF is the coefficient of the face F . It is a framed motive that we will compute in §5.2.7.

Lemma 5.13. Let S ⊂ Hn be triangulated by a finite number of simplices S = PM i=1 ∆i . Suppose B is a compact set contained in the interior of S, and F0 is a face in this triangulation which is contained in B. Then in the reduced coproduct

the coefficient

PM

e ∆

i i=1 cF0

M X i=1

M X X mot(∆i ) = mot(F ) ⊗ ciF , i=1 F

of F0 vanishes.

Proof. The coproduct respects subdivision (5.40), so we can modify the triangulation inside B without affecting the coproduct. But we can find a different triangulation of S in which F0 does not occur at all, so its coefficient must be 0. Another way to see this is in the proof of lemma 5.11. There, the framed motives defined by the complexes (5.41) and (5.42) are equivalent. Since no hyperplane passing through y appears in the latter complex, it cannot occur in the reduced coproduct. The same method of proof gives a similar result for faces passing through a point at infinity x ∈ ∂Hn , including the one-dimensional ones. n

Lemma 5.14. Let x ∈ ∂Hn , and let S ⊂ H be triangulated by a finite number of PM simplices S = i=1 ∆i , where each ∆i has at most one vertex on ∂Hn which is necessarily the point x. Let B be a compact set contained in a horosphere around x, and let Bx denote the open cone over B with vertex at x. Suppose that Bx is contained in the interior of S, and let F0 ⊂ Bx be a face in this triangulation which passes e PM mot(∆i ) . through x. Then the coefficient of F0 vanishes in the coproduct ∆ i=1

Theorem 5.15. Let M be a product-hyperbolic manifold defined over the number fields (k1 , . . . , kN ) as above. Then mot(M ) ∈ A(k1 )⊗. . .⊗A(kN ) defines an element in Ext1MT(k1 ) (Q(0), Q(n1 )) ⊗ . . . ⊗ Ext1MT(kN ) (Q(0), Q(nN )) . QN Proof. Let Xn = i=1 H2ni −1 , and M = Xn /Γ. By corollary 4.12, M admits a PM (i) (i) finite tiling with product simplices i=1 ∆1 × . . . × ∆N defined over (k1 , . . . , kN )


45

with at most one vertex at infinity. This tiling lifts to a Γ-equivariant tiling of Xn . (i) (i) Consider any finite face of ∆1 × . . . × ∆N , which is necessarily of the form (i) ∆j n

E = E1 × . . . × EN ,

for 1 ≤ j ≤ N . Let F = F1 × . . . × FN be a lift of the where Ej is a face of face E in the tiling of X , and let B ⊂ Xn denote a compact ball containing F . By subdividing, we can assume that the tiling is a triangulation in the neighbourhood of B. There exists a finite number of elements γ1 , . . . , γK ∈ Γ, such that S=

K X M X l=1 i=1

(i)

(i)

γl ∆1 × . . . × ∆N

,

completely covers B. The conditions of lemma 5.13 hold in each component, and we conclude that the coefficient of Fj vanishes in the j th component of the coproduct e ⊗ idj+1 ⊗ . . . ⊗ idN (mot(S)) , id1 ⊗ . . . ⊗ idj−1 ⊗ ∆

for each 1 ≤ j ≤ N . Since each γi is an isometry, mot(S) =

K X

mot(γi M ) = Kmot(M ) .

i=1

We conclude that the coefficient of mot(Ej ) in the j th component of the coproduct e id⊗. . .⊗ ∆⊗. . .⊗id(mot(M )) is zero. A similar argument using lemma 5.14 implies that the coefficients of all faces with a vertex at infinity vanish too. It follows that e nN . e n1 ⊗ . . . ⊗ ker ∆ mot(M ) ∈ ker ∆

The statement of the theorem follows on identifying Ext1MT(ki ) (Q(0), Q(ni )) with e ni ker ∆ , by corollary 5.3. An (ki ) i

5.2.5. Another proof of theorem 5.15. Consider the pth graded part of the reduced e p (mot(M )). Let V ⊂ Ap (Q) ⊗ An−p (Q) denote the vector space coproduct ∆ spanned by all the terms which occur in the reduced coproduct: mot(F ) ⊗ cF

where F ranges over all faces in the triangulation of M . Since the triangulation e p (mot(M )) = v ∈ V. Now consider is finite, V is finite dimensional, and we set ∆ two large balls Br ⊂ Br+s of radius r and r + s respectively contained in Xn . Fixing a sufficiently large s, we can find, for all r sufficiently large, γ1 , . . . , γNr P S (i) such that Br ⊂ 1≤i≤Nr γi (D) ⊂ Br+s is a triangulation, where D = M i=1 ∆1 × (i)

n . . . × ∆N is the lift of a tiling for M to X S as above. Consider the complex of hyperplanes which defines the motive mot( i γi (D)). This is framed equivalent to the same complex from which we have removed all internal hyperplanes, by the same argument as in the proof of lemma 5.11. Therefore, the reduced coproduct e p (mot(M )) can be computed by considering all faces F of the triangulation which ∆ lie in between Br and Br+s . By volume considerations, S the number of such faces n is of order r(dim X −1) . On the other hand, we have mot( i γi (M )) = Nr mot(M ), e p (mot(M )) = Nr v where Nr is of order rdim Xn . Thus v must be and therefore, ∆ n n equal to 0, otherwise Nr v would simultaneously be of order r(dim X −1) and rdim X in any given basis of V , which gives a contradiction.

46

FRANCIS C.S. BROWN

5.2.6. The twisted Galois action. It remains to show, in the case when M is equivariant, that its framed motive mot(M ) is a determinant. Assume that all hyperbolic components are of equal odd dimension 2n − 1, i.e., n1 = . . . = nN , and let k be a totally real number field with a set of embeddings Σ = {σ1 , . . . , σN } into R.4 Let ki = σi (k) and suppose that M is equivariant with respect to Σ. Theorem 5.16. Let M be an equivariant product-hyperbolic manifold as above. Then, in the notation of §5.1.4, the framed motive of M is a determinant: VΣ mot(M ) ∈ An (k) . PM (i) (i) (i) Proof. Let M = i=1 ∆1 × . . . × ∆N be a product-tiling for M , where ∆j is defined over kj , and has at most one vertex at infinity. Since the cusps of M are equivariant (lemma 4.5), the image of this tiling under the twisted action of the symmetric group SN is another tiling for π(M ) = M (lemma 4.13): M=

M X i=1

(i)

(i)

−N −1 ∆N ) , for all π ∈ SN , (σ1 σπ(1) ∆π(1) ) × . . . × (σN σπ(1)

It follows that mot(M ) and mot(π(M )) are equivalent up to a sign, determined by ∨ the action of π on the framings. First, π preserves the framing in grW 0 mot(M ) corresponding to the fundamental class of M , precisely because M is equivariant. The n framing in grW 2n mot(M ) corresponds to the volume form on X , which is alternating: π[ωQ1 ∧ . . . ∧ ωQN ] = ε(π)[ωQ1 ∧ . . . ∧ ωQN ] ,

for all π ∈ SN .

This is because each form ωQi is of odd degree, since each hyperbolic component of Xn is, by assumption, of odd dimension. In conclusion, the framed motive of M is alternating under this action of SN , i.e., mot(M ) = ε(π) π(mot(M ))

for all π ∈ SN .

Thus mot(M ) ∈ An1 (σ1 (k)) ⊗ . . . ⊗ An1 (σN (k)) is a determinant in the sense of §5.1.4. m

5.2.7. Quotient motives and spherical angles. Let m = 2n − 1, and let ∆ ⊂ HQ denote a geodesic simplex with at most one vertex at infinity. As above, it defines a set of hyperplanes L0 , . . . , Lm in general position with respect to a smooth quadric Q. We wish to compute the coefficient motive cF of an odd-dimensional face F of ∆. It corresponds to a hyperplane LJ = Lj1 ∩ . . . ∩ Lj2r , and hence a complex a a LI ∩ Q ⇇ LJ ∩ Q . LI ∩ Q ⇇ . . . ⇇ (5.45) Q← |I|=1, I⊂J

|I|=2r−1, I⊂J

This defines a mixed Tate motive which we denote by [ (5.46) h(∆F ) = H 2n−2 (Q, Lj ∩ Q) . j∈J

Recall that if Q0 is a smooth quadric of even dimension 2d, then H 2d (Q0 ) ∼ = Q(−d) ⊕ Q(−d). The difference of the two classes defines, up to sign, a map vQ0 : Q(−d) −→ H 2d (Q0 ) .

4In the exceptional case n = 2, if we identify SO+ (3, 1) with PSL (C) we can allow k be to be 2 any number field L and σi to be complex places of L.


47

2n−2 One verifies that grW (Q), which is framed by vQ , and that 2n−2 h(∆F ) = H W gr2n−2r−2 h(∆F ) ∼ = Q(r + 1 − n), which is framed by the dual of vQ∩LJ . In the Hodge realisation, these framings correspond to the volume form on Q:

(5.47)

vQ = ±in

2n X p c i ∧ . . . ∧ dx2n−1 , (−1)i xi dx0 ∧ . . . ∧ dx det q i=1

and the relative homology class of the spherical simplex SF ⊂ Q(R) cut out by the hyperplanes (Lj ∩ Q)j∈J on the set of real points of the quadric Q. The sign is determined by the orientations. See §5.3 below for an explicit example. Lemma 5.17. Let ∆ be as above. For every strict odd-dimensional subface F of ∆, the coefficient cF of F is framed equivalent to the twisted motive h(∆F )(1). Proof. We assume that ∆ has a single vertex x at infinity, and number the hyperplanes L0 , . . . , Lm in such a way that L0 does not meet x, but L1 , . . . , Lm do. The face F corresponds to a subset J ⊂ {0, 1, . . . , m}. If F meets x then set Je = J ∪ {−1}, otherwise set Je = J, and consider the complex: a a e2n−1 \Q e← e I \Q eI ⇇ . . . ⇇ e I \Q eI ⇇ L e J \Q eJ . (5.48) P L L |I|=1, I⊂Je

|I|=2r−1, I⊂Je

e −1 is the exceptional divisor. There is an obvious map from (5.23) Recall that L to (5.48). Notice that the set of divisors Q and {Lj }j∈J are now normal crossing in all cases. Blowing the exceptional divisor back down gives a further map from (5.48) to the complex a a LI \QI ⇇ LJ \QJ . LI \QI ⇇ . . . ⇇ (5.49) P2n−1 \Q ← |I|=1, I⊂J

|I|=2r−1, I⊂J

Finally, taking the residue along the quadric Q gives a map to (5.45). This defines the required morphism φ : h(∆) −→ h(∆F )(1), and it is an exercise to check that the framings are compatible. Lemma 5.18. The maximal period of the quotient motive cF is Z 1.3 . . . (2n − 3) vol(SF ) , vQ = 2πi i1−n (5.50) 2πi 2n SF

where SF is the spherical simplex defined earlier, and vol(SF ) its spherical volume.

We deduce from (5.32) that there is a single relation between the dihedral angles of all faces F which meet the point x at infinity. This corresponds to the fact that there is a single relation between the angles of the Euclidean simplex at infinity ∆∞ since it has an even number of dimensions. 5.2.8. Volume and the main theorem. The volume of a hyperbolic simplex with at most one vertex at infinity is determined by its real period. The proof is identical to Goncharov’s proof of the same result for finite simplices [29]. Corollary 5.19. Let k be a number field and let σ : k → R be a real embedding. Let ∆ denote an oriented hyperbolic geodesic simplex which is defined over σ(k) and has at most one vertex at infinity. Then vol(∆) = (2π)n Rσ (mot(∆)) .

48

FRANCIS C.S. BROWN

Proof. Let Pσ be a period matrix for mot(∆). Its first column is given by integrating the form ωQe over a basis for the homology of h(∆). The previous lemma proves −n that, in the basis given by the e∨ R except for the first, which e , all entries lie in i F n−1 −n is given by i vol(∆). Thus, up to signs, Im ((2iπ) Pσ v0 ) is the column vector ((2π)−n vol(∆), 0, . . . , 0), and it follows from the definition and the choice of signs on the framings that Rσ [h(∆), ωQe , ∆] = hIm ((2iπ)−n Pσ v0 ), Re (Pσ∗ fn )i which is exactly (2π)−n vol(∆) × 1. Putting the various elements together, we obtain the following theorem. Theorem 5.20. Let M be a product-hyperbolic manifold modelled on H2n1 −1 × . . . × H2nN −1 , and defined over the fields (k1 , . . . , kN ). Then the framed motive of M is a well-defined element: mot(M ) ∈ Ext1MT(k1 ) (Q(0), Q(n1 )) ⊗ . . . ⊗ Ext1MT(kN ) (Q(0), Q(nN )) , such that the volume of M is given by the Hodge regulator: vol(M ) = (2π)n1 +...+nN RΣ (mot(M )) . If M is equivariant with respect to Σ = {σ1 , . . . , σN }, and ni = n for all i, then VΣ Ext1MT(k) (Q(0), Q(n)) . mot(M ) ∈

5.2.9. Remark on generalised Dehn invariants. An alternative way to prove the previous theorem is using Dehn invariants, along the lines of [29]. Let M n = Xn /Γ be a product-hyperbolic manifold defined over the fields S = (k1 , . . . , kN ). By §4.7, M defines a scissors-congruence class ξM ∈ P(Xn ). The map φ : P(Hn ) −→ A(Q) which to any finite hyperbolic simplex ∆N defined over Q associates the framed motive mot(∆) gives a map φ : P(Xn ) −→ 1≤i≤N Ani (Q) . We can set mot(M ) = φ(ξM ) .

It follows from lemma 5.11 that this coincides with definition 5.12. Proposition 5.21. [29]. The map φ takes the kernel of the Dehn invariant to the kernel of the reduced coproduct: en ∼ φ : ker D2n−1 −→ ker ∆ = Ext1MT(Q) (Q(0), Q(n)) .

By theorem 4.17, we have Dn (ξM ) = 0. By the definition 4.16 of the generalised Dehn invariant, ξM ∈ ker(D2n1 −1 ) ⊗ . . . ⊗ ker(D2nN −1 ), and therefore e nN ) ∼ e n1 ) ⊗ . . . ⊗ ker(∆ φ(ξM ) ∈ ker(∆ =

N O

Ext1MT(ki ) (Q(0), Q(n)) .

i=1

The fact that φ(ξM ) is a determinant in the equivariant case follows from theorem 4.17 in the same way as the proof of theorem 5.16. 5.3. Examples. We illustrate the previous constructions with explicit computations of the motives of hyperbolic simplices in small dimensions.


49

5.3.1. A hyperbolic line element. First consider the simplest case of a hyperbolic line segment L in H1 ∼ = R. After complexifying, we obtain a pair of points {x0 , x1 } ∈ P1 , and a quadric Q ⊂ P1 which consists of a pair of points {q0 , q1 } ∈ P1 . Then

h(L) = H 1 (P1 \{q0 , q1 }, {x0 , x1 }) , ∼ which is just a Kummer motive, i.e., grW • h(L) = Q(0) ⊕ Q(−1). Its maximal period is given by the logarithm of the cross-ratio, which is a projective invariant: Z x1 (x − q )(x − q ) 1 dx dx 1 1 1 0 0 1 = log [x1 x0 q0 q1 ] . = log − 2 x − q x − q 2 (x − q )(x − q ) 2 0 1 0 0 1 1 x0 The real period of h(L) is half the real part of the logarithm log [x1 x0 |q0 , q1 ] , which, up to a sign, is the hyperbolic length of the oriented line segment {x0 , x1 }. (5.51)

5.3.2. Hyperbolic triangles in the hyperbolic plane. Now consider a finite triangle T in H2 . It defines a set of 3 lines L0 , L1 , L2 and a smooth quadric Q in P2 . Then [ h(T ) = H 2 P2 \Q, Li \(Li ∩ Q) . 0≤i≤2

3 W From (5.21), we have grW 2 h(T ) = Q(−1) , and gr0 h(T ) = Q(0). Each side of T defines a motive (5.51) whose period is its hyperbolic length. It follows that h(T ) splits as a direct sum of Kummer motives, one corresponding to each side, and (the dual of) its period matrix can be written:   1 0 0 0  2ℓ0 2iπ 0 0     2ℓ1 0 2iπ 0  2ℓ2 0 0 2iπ

where ℓ0 , ℓ1 , ℓ2 are the hyperbolic lengths of its sides. Note that the triple {ℓ0 , ℓ1 , ℓ2 } is a complete isometry invariant for T . Since there is no non-zero framing in grW 4 h(T ), the framed motive mot(T ) is 0 in this case. Now consider what happens when one vertex x = L1 ∩L2 of T is on the absolute. e2 ∼ After blowing up the point x, we obtain a smooth quadric Q in P = P1 × P1 , and e e e e a configuration of four lines L−1 , L0 , L1 , L2 as shown below (left). Now, [ e 2 \Q, e ei) , h(T ) = H 2 (P L −1≤i≤2

2 W ∼ and (5.34) gives grW 2 h(T ) = Q(−1) , gr0 h(T ) = Q(0). It is therefore the direct sum of two Kummer motives. One of them comes from the inclusion of the side L0 of finite length, whose period is its hyperbolic length. To compute the period of the other, consider the following diagram. Note that the motive h(T ) is uniquely

e −1 L

e1 L

x

e Q

e0 L

e2 L

p

R

s q

ℓe2

ℓe1

r 0

t1

t2

1

50

FRANCIS C.S. BROWN

determined by the five points x, p, q, r, s ∈ ∂H2 ∼ = P1 (R). Lemma 5.22. The periods of h(T ) are given by the two quantities (x − q)2 (p − r)(r − s) (p − r)(q − s) 1 1 and ℓ = . log (5.52) ℓx = log 0 2 (x − r)2 (p − q)(q − s) 2 (p − q)(r − s)

Proof. The periods of h(T ) define a Kummer variation on the configuration space of 5 distinct points in P1 (R) modulo the action of PSL2 (R). This is the moduli space M0,5 (R) of genus 0 curves with 5 marked points. By projective transformation, set p = 0, q = t1 , r = t2 , s = 1, x = ∞. The space of logarithms on M0,5 is spanned by log(t1 ), log(t2 ), log(1 − t1 ), log(1 − t2 ), log(t2 − t1 ). The periods of h(T ) are additive with respect to subdivision (the dotted line above). A simple calculation shows that the vector space of additive functions is spanned by the two functions: t (1 − t ) t (1 − t ) 2 2 2 1 2 ℓx = log and 2 ℓ0 = log . t1 (1 − t1 ) t1 (1 − t2 )

Rewriting t1 , t2 , 1 − t1 , 1 − t2 as cross-ratios, we obtain formula (5.52).

The quantity ℓx (resp. ℓ0 ) is anti-invariant (resp. invariant) under the transformation (p, q) ↔ (r, s). One checks that ℓ0 is the hyperbolic length of the line segment L0 , and that ℓx is the difference of the regularised lengths ℓe1 − ℓe2 of the sides L1 , L2 . Here ℓei is defined to be the length of the truncated line segment of side Li up to a horoball neighbourhood of x (depicted by a horizontal dotted line at height R in the above diagram (right)). The quantity ℓe1 − ℓe2 is independent of the choice of horoball. In conclusion, a (dual) period matrix for h(T ) is:   1 0 0  2ℓ0 2iπ 0  2ℓx 0 2iπ 5.3.3. A finite simplex in hyperbolic 3 space. Consider a finite hyperbolic geodesic simplex ∆ in H3 , which is given by four hyperplanes L0 , . . . , L3 in general position relative to a smooth quadric Q in P3 . From (5.21), we have grW h(∆) ∼ grW h(∆) ∼ = Q(0) . = Q(−1)6 , grW h(∆) ∼ = Q(−2) , 0

2

4

To compute the periods, and to see why the Dehn invariant naturally appears in the coproduct, we look at the skelton and coskelton construction for the complex which defines the motive h(∆). Therefore, consider the simplicial scheme (5.16): a a a (5.53) P3 \Q ← Li \Qi ⇇ Lij \Qij ⇇ Lijk , 0≤i≤3

0≤i

Dedekind zeta motives for totally real fields

Dedekind zeta motives for totally real fields

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