On the growth of torsion in the cohomology of arithmetic groups
aa r X i v : . [ m a t h . N T ] F e b ON THE GROWTH OF TORSION IN THE COHOMOLOGY OFARITHMETIC GROUPS
WERNER M ¨ULLER AND JONATHAN PFAFF
Abstract.
In this paper we consider certain families of arithmetic subgroups of SO ( p, q )and SL ( R ), respectively. We study the cohomology of such arithmetic groups with co-efficients in arithmetically defined modules. We show that for natural sequences of suchmodules the torsion in the cohomology grows exponentially. Introduction
Let G be a semi-simple connected algebraic group over Q , K a maximal compact sub-group of its group of real points G ( R ). Let e X = G ( R ) /K be the associated Riemann-ian symmetric space. Let g and k be the Lie algebras of G ( R ) and K , respectively.Put δ ( e X ) = rank( g C ) − rank( k C ). Sometimes δ ( e X ) is called the fundamental rank. LetΓ ⊂ G ( Q ) be an arithmetic subgroup and X = Γ \ e X the corresponding locally symmetricspace. We assume that G is anisotropic over Q , which implies that Γ is cocompact in G ( R ).Let M be an arithmetic Γ-module, which means that M is a finite rank free Z -module, andthere exists an algebraic representation of G on M ⊗ Z Q such that Γ preserves M . Thenthe cohomology H ∗ (Γ , M ) is a finite rank Z -module. Note that if Γ is torsion free, then H ∗ (Γ , M ) ∼ = H ∗ (Γ \ e X, M ), where M is the local system of free Z -modules associated to M .For arithmetic reasons, one expects that if δ ( e X ) = 0, there is little torsion in H ∗ (Γ , M )and the free part dominates the cohomology. On the other hand, if δ ( e X ) = 1, one expects alot of torsion in the cohomology and the free part to be small. This has been substantiatedby Bergeron and Venkatesh [BV], who studied the growth of the torsion if Γ varies through asequence of congrunce subgroups Γ n for which the injectivity radius of Γ n \ ˜ X goes to infinity.They showed that if δ ( ˜ X ) = 1 and M is strongly acylic, the torsion grows exponentiallyproportional to the volume of Γ n \ ˜ X . Furthermore, for compact oriented hyperbolic 3-manifolds, in [MaM¨u] the growth of the torsion has been studied if Γ is fixed but theΓ-module M grows. More precisely, let X = Γ \ H be a compact, oriented hyperbolic3-manifold with Γ ⊂ SL(2 , C ). Let V m be the holomorphic irreducible representation ofSL(2 , C ) of dimension m + 1. By [BW] one has H ∗ (Γ , V m ) = 0. It was proved in [MaM¨u]that for each even k ∈ N there exists a Γ-invariant lattice M k ⊂ V k . Then H p (Γ , M k ) is a Date : October 1, 2018.1991
Mathematics Subject Classification.
Primary: 11F75.
Key words and phrases.
Arithmetic groups, cohomology. finite abelian group for all p , and the main result of [MaM¨u] is the following asymptoticformula(1.1) lim k →∞ log | H (Γ , M k ) | k = 2 π vol( X ) , and the estimation(1.2) log | H p (Γ , M k ) | ≪ k log k, p = 1 , . Note that H (Γ , M k ) = 0.The goal of the present paper is to study the growth of the torsion if M varies, for allcompact arithmetic quotients Γ \ ˜ X of irreducible symmetric spaces ˜ X with δ ( ˜ X ) = 1. Bythe classification of simple Lie groups, the irreducible symmetric spaces with δ ( ˜ X ) = 1 are e X = SO ( p, q ) / SO( p ) × SO( q ), for p, q odd, and e X = SL(3 , R ) / SO(3).The first family of arithmetic groups that we consider are cocompact arithmetic sub-groups of SO ( p, q ) that arise from quadratic forms over totally real number fields. Moreprecisely, let F be a totally real finite Galois extension of Q of degree d >
1. We fix anembedding F ⊂ R . Let Q : R p + q → R be a non-degenerate quadratic form defined over F of signature ( p, q ). Assume that all non-trivial Galois conjugates of Q are positive definite.Let G := SO Q ⊂ GL p + q be the special orthogonal group of Q , i.e., the subgroup of allelements of determinant one leaving Q invariant. This is a connected algebraic group over F and its group of real points G ( R ) is isomorphic to SO( p, q ).Let O F be the ring of algebraic integers of F , and let G O F be the group of O F -valuedpoints of G . Then G O F is a discrete cocompact subgroup of G ( R ). Via the isomorphism G ( R ) ∼ = SO( p, q ), it corresponds to a discrete cocompact subgroup Γ of SO( p, q ) (seesection 3). If we pass to an appropriate subgroup of finite index Γ ⊂ Γ , we may assumethat Γ is torsion free and that it is containd in SO ( p, q ).Since G is only defined over F , we need to generalize the notion of an arithmetic Γ-module. Let G ′ = Res F/ Q ( G ) be the algebraic Q -group obtained from G by restrictionof scalars. Let ∆ : G → G ′ be the diagonal embedding. Consider Γ as a subgroup of G O F . Let Γ ′ = ∆(Γ). Then Γ ′ ⊂ G ′ ( Q ) is an arithmetic subgroup. Let M be an arithmeticΓ ′ -module. Since Γ ∼ = Γ ′ , it becomes a Γ-module and we also call it an arithmetic Γ-module.To state our main result, we need to introduce some notation. Let e X d be the compactdual symmetric space of e X . We chose an SO ( p, q )-invariant metric on e X and equip X and e X d with the induced metrics. Assume that p, q are odd, p ≥ q , p > n := ( p + q ) / −
1. We let ǫ ( q ) := 0 for q = 1 and ǫ ( q ) := 1 for q > C p,q = ( − pq − ǫ ( q ) π vol( e X d ) (cid:18) n p − (cid:19) . Then our first main result is the following theorem.
Theorem 1.1.
Let F be a totally real Galois extension of Q of degree d > . Let Γ be atorsion free cocompact arithmetic subgroup of SO ( p, q ) derived from a quadratic form Q over F as above. Then there exists a sequence of arithmetic Γ -modules M m , m ∈ N , with the following properties. The rank rk Z ( M m ) is a polynomial in m and there exists C > ,which depends only on n , such that rk Z ( M m ) = C d m dn ( n +1) / + O ( m dn ( n +1) / − ) as m → ∞ . Furthermore each cohomology group H j (Γ , M m ) is finite and X j ≥ ( − j log (cid:12)(cid:12) H j (Γ , M m ) (cid:12)(cid:12) = − C p,q vol(Γ \ e X ) m rk Z ( M m ) + O (rk Z ( M m )) as m → ∞ . Let k = (dim( e X ) + 1) /
2. Then it follows from Theorem 1.1 that there exists a constant˜ C p,q >
0, which depends only on p, q , such that(1.4) lim inf m X j ≡ k (2) log | H j (Γ , M m ) | m dn ( n +1) / ≥ ˜ C p,q d vol(Γ \ e X ) . Thus there is at least one j for which | H j (Γ , M m ) | grows exponentially in m . Given (1.1)and (1.2), one is tempted to pose the following conjecture. Conjecture . Let Γ and M m , m ∈ N , be as above. Thenlim m →∞ log | H j (Γ , M m ) | m dn ( n +1) / = ( ˜ C p,q d vol(Γ \ e X ) , j = (dim( e X ) + 1) / , , else . The next case that we consider are arithmetic subgroups of SL ( R ) which arise from9-dimensional division algebras D over Q . Let o be an order in D . Then o induces anarithmetic subgroup Γ of SL ( R ) which is cocompact (see section 4). After passing to asubgroup of finite index, we may assume that Γ is torsion-free.Let t C be the standard complexified Cartan-subalgebra of the Lie algebra of SL ( R )equipped with the standard ordering of the roots and let ω , ω ∈ t ∗ C be the correspondingfundamental weights, see (4.30). If Λ = τ ω + τ ω ∈ t ∗ C , τ , τ ∈ N is a dominant weightand τ Λ is the corresponding irreducible finite-dimensional representation of SL ( R ), we letΛ θ be the highest weight of τ Λ ◦ θ . One has Λ θ = τ ω + τ ω . Moreover, for m ∈ N we let τ Λ ( m ) be the irreducible representation of SL(3 , R ) on a complex vector space V Λ ( m ) withhighest weight m Λ. We regard V Λ ( m ) as a real vector-space. Let e X = SL(3 , R ) / SO(3),let e X d be the compact dual of e X and let X = Γ \ e X . We fix a G -invariant metric on e X which induces metrics on X and on e X d . Then our main result for the SL ( R )-case is thefollowing theorem. Theorem 1.3.
Let Γ be an arithmetic subgroup of SL ( R ) which arises from a 9-dimensionaldivision algebra over Q . Let Λ ∈ t ∗ C be a highest weight with Λ θ = Λ . Then for each m there exists a lattice M Λ ( m ) in V Λ ( m ) which is stable under Γ . Moreover, each cohomology WERNER M ¨ULLER AND JONATHAN PFAFF group H p (Γ , M Λ ( m )) is finite and one has X p =0 ( − p log | H p (Γ , M Λ ( m )) | = − π vol( X )vol( e X d ) C (Λ) m · rk Z M Λ ( m ) + O (rk Z M Λ ( m )) , (1.5) as m → ∞ , where C (Λ) > is a constant which depends only on Λ . If Λ equals one of thefundamental weights ω f then C (Λ) = 4 / . The rank of M Λ ( m ) can be computed explicitly as follows. Firstly, if Λ is equal to oneof the fundamental weights ω or ω , then Λ θ = Λ and Weyl’s dimension formula givesrk Z M Λ ( m ) = dim R ( V Λ ( m )) = m O ( m ) , as m → ∞ . Secondly, if Λ = τ ω + τ ω , τ , τ ∈ N , τ τ = 0, then the condition Λ = Λ θ is equivalent to τ = τ and again by Weyl’s dimension formula one hasrk Z M Λ ( m ) = dim R ( V Λ ( m )) = τ τ + τ τ m + O ( m ) , as m → ∞ . Let M i,m := M ω i ( m ), i = 1 ,
2. Then it follows that(1.6) lim inf m X j =0 log | H j +1 (Γ , M i,m ) | m ≥ π e X d ) vol( X ) . Again, by (1.1) and (1.2), one is led to the following conjecture.
Conjecture . Let Γ and M i,m , m ∈ N , be as above. Then one has(1.7) lim m →∞ log | H (Γ , M i,m ) | m = 2 π e X d ) vol( X )and(1.8) log | H j (Γ , M i,m ) | = o ( m ) , j = 3 . There are similar statements for highest weights Λ = τ ω + τ ω with τ τ = 0.Next we describe our approach to prove the main results. As in [MaM¨u], it is basedon the study of the analytic torsion. To begin with we consider an arbitrary connectedsemi-simple algebraic group G over Q . Let the notation be as at the beginning of theintroduction. Assume that δ ( e X ) = 1. Choose a G -invariant Riemannian metric g on e X .Assume that Γ ⊂ G ( Q ) is torsion free. Let X := Γ \ e X equipped with the metric inducedby G . Then we have H ∗ (Γ , M ) = H ∗ ( X, M ). Let V := M ⊗ Z R and let ρ : G ( R ) → GL( V )be the representation associated to the arithmetic Γ-module M . Let E → X be the flat vector bundle associated to ρ | Γ . Choose a Hermitian fibre metric in E . Let T X ( ρ ) ∈ R + be the analytic torsion of X and E . Recall that(1.9) log T X ( ρ ) = 12 n X p =1 ( − p p dds ζ p ( s ; ρ ) (cid:12)(cid:12) s =0 , where ζ p ( s ; ρ ) is the zeta function of the Laplace operator ∆ p ( ρ ) on E -valued p -forms and n = dim X (see [M¨u1]). Assume that ρ is acyclic, that is H ∗ ( X, E ) = 0. Then T X ( ρ ) ismetric independent [M¨u1, Corollary 2.7] and equals the Reidemeister torsion τ X ( ρ ) [M¨u1,Theorem 1]. Moreover, H ∗ (Γ , M ) is a finite abelian group. Using Proposition 2.1, we get(1.10) log T X ( ρ ) = n X q =0 ( − q +1 log | H q (Γ , M ) | . This is the key equality which we apply to prove Theorems 1.1 and 1.3. In [MP] we studiedthe asymptotic behavior of T X ( τ m ) for certain sequences of irreducible representations of G ( R ) . We will apply the results of [MP] to our case. The main issue is to constructappropriate arithmetic Γ-modules.We start with the case of e X = SO ( p, q ) / SO( p ) × SO( q ), p, q odd. Let G = SO Q bethe special orthogonal group of a quadratic form Q over a totlally real number field F asdefined above. Then G is a connected algebraic group over F . Let G ′ = Res F/ Q ( G ) bethe algebraic Q -group obtained from G by restriction of scalars [Wei]. The group of realpoints G ′ ( R ) is given by G ′ ( R ) ∼ = SO( p, q ) × K , where K is the product of d − p + q ). Then we construct a sequence ρ ( m ) : G ′ → GL ( V m ), m ∈ N , of Q -rational representations such that the irreduciblecomponents of ρ ( m )( R ) : G ′ ( R ) → GL( V m ⊗ Q R ) are of the form considered in [MP,Theorem 1.1]. Let ∆ : G → G ′ be the diagonal embedding. Let Γ ′ = ∆(Γ). Then Γ ′ isan arithmetic subgroup of G ′ ( Q ). Therefore, V m contains a lattice M m which is invariantunder ρ ( m )(Γ ′ ). Through the isomorphism Γ ∼ = Γ ′ , M m becomes a Γ-module. This is ourarithmetic Γ-module. By construction we have H ∗ (Γ , M m ) ∼ = H ∗ (Γ ′ , M m ). Thus it sufficesto prove the statement of Theorem 1.1 for Γ ′ .Let K ′ = SO( p ) × SO( q ) × K . Then K ′ is a maximal compact subgroup of G ′ ( R ) . Let e X ′ = G ′ ( R ) /K ′ and X ′ := Γ ′ \ e X ′ . Now we apply [MP, Propositions 1.2, 1.3] to determinethe asymptotic behavior of T X ′ ( ρ ( m )) as m → ∞ . Finally we use (1.10) to establishTheorem 1.1.The proof of Theorem 1.3 uses similar arguments, which are also based on (1.10) and[MP].The paper is organized as follows. In section 2 we collect some facts about cohomology offundamental groups of manifolds with coefficients in a free Z -module. We also recall someelementary facts about algebraic groups. In section 3 we consider arithmetic subgroups ofSO ( p, q ) and prove Theorem 1.1. The prove of Theorem 1.3 is the content of the finalsection 4. WERNER M ¨ULLER AND JONATHAN PFAFF preliminaries Let X be a closed connected smooth manifold of dimension d . Let Γ := π ( X, x ) be thefundamental group of X with respect to some base point x and let e X be the correpsondinguniversal covering. Thus Γ acts properly discontinuously and fixed point free on e X and X = Γ \ e X . Assume that e X is contractible.Let M be a free finite-rank Z -module and let ρ be a representation of Γ on M . Let H q (Γ , M ) be the q -th cohomology group of Γ with coefficients in M , see [Br]. Thesegroups can be computed as follows. Let L be a smooth triangulation of X and let e L be thelift of L to a triangulation of e X . Let C q ( e L ; Z ) be the free abelian group generated by the q -chains of e L , let C q ( e L ; Z ) := Hom Z ( C q ( e L, Z ); Z ) and let C ∗ ( e L ; Z ) resp. C ∗ ( e L ; Z ) be theassociated simplical chain- resp. cochain complexes. Each C q ( e L ; Z ) is a free Z [Γ] moduleand if one fixes an embedding of L into e L , then the q -cells of L form a base of C q ( e L ; Z )over Z [Γ]. Let C q ( L, M ) := C q ( e L ; Z ) ⊗ Z [Γ] M. Then the C q ( L, M ) form again a cochain complex C ∗ ( L, M ) and the corresponding coho-mology groups will be denoted by H q ( L, M ) . There is an isomorphism C q ( L, M ) ∼ = Hom Z [Γ] ( C q ( e L ; Z ) , M ) , which induces an isomorphism of the corresponding cochain complexes. Since e L is con-tractible, the complex C ∗ ( e L ) is a free resolution of Z over Z [Γ] and thus one has(2.11) H q (Γ , M ) ∼ = H q (Hom Z [Γ] ( C ∗ ( e L ) , M )) ∼ = H q ( L, M ) . Each cohomology group H q ( L, M ) is a finitely generated abelian group. Let H q (Γ , M ) tors be the torsion subgroup of H q (Γ , M ) and let H q (Γ , M ) free = H q (Γ , M ) /H q (Γ , M ) tors be thefree part. Then one has H q (Γ , M ) = H q (Γ , M ) free ⊕ H q (Γ , M ) tors . Now let V := M ⊗ Z C and V R := M ⊗ Z R . Then V is a finite-dimensional complexvector space, V R ⊂ V is a real structure on V and M is a lattice in V R . We regard ρ as arepresentation of Γ on V . Then ρ is unimodular, i.e., | det ρ ( γ ) | = 1 for all γ ∈ Γ. Let C q ( L, V ) := C q ( e L ) ⊗ Z [Γ] V. The C q ( L, V )’s form a chain complex C ∗ ( L, V ) of finite-dimensional C -vector spaces andone has C q ( L, V ) = C q ( L, M ) ⊗ Z C . (2.12)Let E := ˜ X × ρ V be the flat vector bundle over X associated to ρ . Then by the de Rhamisomorphism, the cohomology groups H q ( L, V ) of the complex C ∗ ( L, V ) are canonicallyisomorphic to the cohomology groups H q ( X, E ) of the complex of E -valued differentialforms on X . By Hodge theory they are canonically isomorphic to the space of E -valuedharmonic forms for any choice of metrics on X and E , respectively. We assume that the bundle E is acyclic i.e. that H q ( L ; V ) = H q ( X ; E ) = 0 for all q . This holds in all cases that we study in this paper. Let σ qj , j = 1 , . . . , r q , be theoriented q -simplices of L considered as a preferred basis of the Z [Γ]-module C q ( e L ; Z ). Let e , . . . , e m be a basis of M . Then { σ qj ⊗ e k : j = 1 , . . . , r q , k = 1 , . . . , m } is a preferredbasis of C q ( L ; M ) and also of C q ( L ; V ). Let τ X ( ρ ) ∈ R + be the Reidemeister torsion withrespect to these volume elements (see [M¨u1], [MaM¨u]). Note that τ X ( ρ ) = | τ C X ( ρ ) | , where τ C X ( ρ ) ∈ C × is the complex Reidemeister torsion. Since ρ is acyclic, τ X ( ρ ) is a combinatorialinvariant of X and ρ which is independent of the choices that we made (see [M¨u1, section1]). Moreover, each cohomology group H q (Γ , M ) is finite, i.e., H q (Γ , M ) = H q (Γ , M ) tors and the order | H q (Γ , M ) | of these groups is related to the Reidemeister torsion as follows. Proposition 2.1.
Assume that H q ( X, E ) = 0 for all q . Then one has d X q =0 ( − q +1 log | H q (Γ , M ) | = log τ X ( ρ ) . Proof.
Let C ∗ ( L, V R ) be the chain complex of the finite-dimensional real vector spaces C q ( L, V R ) := C q ( e L ) ⊗ Z [Γ] V R , q = 0 , ..., d. We have C q ( L, V R ) = C q ( L, M ) ⊗ Z R . Let ρ R : Γ → GL( V R ) be the representation induced by ρ and let E R := e X × ρ R V R be theassociated flat real vector bundle. Then H ∗ ( X ; E R ) = 0. The basis of the free Z -module C q ( L ; M ), described above, gives rise to a distinguished basis of C q ( L ; V R ). Let τ X ( ρ R ) bethe Reidemeister torsion of the complex C ∗ ( L ; V R ) with respect to volume elements definedby these bases. Then it follows from (2.11) and (2.12) as in [BV, section 2.2] that(2.13) log τ X ( ρ R ) = d X q =0 ( − q +1 log | H q (Γ , M ) | . See also [MaM¨u, Proposition 2.3] and [Tu, Lemma 2.1.1]. Since the coboundary opera-tors of the complexes C ∗ ( L ; V R ) and C ∗ ( L ; V ), respectively, are induced by the cobound-ary operators of C ∗ ( L ; M ), it follows from the definition of the Reidemeister torsion that τ X ( ρ R ) = τ X ( ρ ). Combined with (2.13) the proposition follows. (cid:3) Finally we recall some facts conerning linear algebraic groups. For all details we referto [Bo2]. Let F be a finite Galois extension of Q with Galois group Σ := Gal( F/ Q ). For σ ∈ Σ and x ∈ F let x σ denote the image of x under σ . If G is a linear algebraic groupover F with coordinate algebra F [ G ], let G σ denote the linear algebraic group conjugateby σ , see [Bo2]. If G is realized as the zero set in some F n of an ideal I in F [ X , . . . , X n ],then G σ is the zero set of the ideal I σ , where I σ is obtained from I by applying σ to eachpolynomial coefficient. Each F -rational homomorphism ρ : G → H of linear algebraicgroups over F induces canonically an F -rational homomorphism ρ σ : G σ → H σ . WERNER M ¨ULLER AND JONATHAN PFAFF If G is an algebraic group defined over F , an algebraic group G ′ defined over Q togetherwith an F -rational isomorphism µ : G ′ × Q F → G is called a Q -structure of G . The Q -structure canonically induces an action of Σ on the coordinate algebra of G and thus on G itself.Let V be a finite-dimensional F -vector space. A Q -structure V of V is a Q -subspace V of V such that the embedding V ֒ → V induces an isomorphism V ⊗ Q F of F -vectorspaces. For σ ∈ Σ a Q -linear automorphism A of V is called σ -linear if A ( λv ) = σ ( λ ) A ( v ), λ ∈ F , v ∈ V . Then a semi-linear action of Σ on V is given by a family { f σ } σ ∈ Σ of σ -linear automorphisms f σ of V , satisfying f στ = f σ ◦ f τ , σ, τ ∈ Σ. Given a semi-linearaction of Σ on V , the set V Σ := { v ∈ V : f σ ( v ) = v, ∀ σ ∈ Σ } is a Q -structure of V andevery Q -structure is of this form (see [Bo2, AG.14.2]). If V is a Q -structure of V , thenGL( V ) is a Q -structure of GL( V ) and the corresponding action of Σ on GL( V ) is givenby σ · g := f σ ◦ g ◦ f − σ , g ∈ GL( V ).3. Arithmetic subgroups of SO ( p, q )Let p, q ∈ N be odd. Put p = ( p − / , q = ( q − / , n := p + q . We denote by SO( p, q ) the group of isometries of the standard quadratic form of signature( p, q ) on R p + q with determinant 1. Let SO ( p, q ) denote the identity component of SO( p, q ).The group SO ( p, q ) is of fundamental rank one. Let g be the Lie algebra of SO ( p, q ). Wechoose the fundamental Cartan subalgebra as follows. Let E i,j be the ( p + q ) × ( p + q )-matrixwhich is one at the i -th row and j -th column and which is zero elsewhere. Let H := E p,p +1 + E p +1 ,p . (3.14)and let(3.15) H i := ( √− E i − , i − − E i − , i − ) , ≤ i ≤ p + 1 √− E i − , i − E i, i − ) p + 1 < i ≤ n + 1 . Then h := H ⊕ n +1 M i =2 √− H i is a Cartan subalgebra of g . Define e i ∈ h ∗ C , i = 1 , . . . , n + 1, by e i ( H j ) = δ i,j , ≤ i, j ≤ n + 1 . The finite-dimensional irreducible complex representations τ of SO ( p, q ) are parametrizedby their highest weights Λ( τ ) ∈ h ∗ C given byΛ( τ ) = k ( τ ) e + · · · + k n +1 ( τ ) e n +1 , ( k ( τ ) , . . . k n +1 ( τ )) ∈ Z n +1 ,k ( τ ) ≥ k ( τ ) ≥ · · · ≥ k n ( τ ) ≥ | k n +1 ( τ ) | . (3.16) For Λ( τ ) a weight as in (3.16), the highest weight Λ( τ θ ) of the representation τ ◦ θ isΛ( τ θ ) = k ( τ ) e + · · · + k n ( τ ) e n − k n +1 ( τ ) e n +1 . (3.17)If we let ω + f,n := n +1 X j =1 e j ; ω − f,n := ( ω + f,n ) θ = n X j =1 e j − e n +1 , (3.18)then ω ± f,n are the fundamental weights which are not invariant under θ . We now recallthe construction of certain arithmetically defined cocompact subgroups of SO ( p, q ). Formore details see [Sch, section 3.2, Appendix B] and for the SO ( p, F be a totally real number field of degree d = [ F : Q ] >
1. Let Σ be the Galoisgroup of F over Q . We fix an embedding F ⊂ R . Let 1 ∈ Σ be the identity. Let α j ∈ F ∗ , j = 1 , . . . , p + q , be such thatsign( α j ) = ( +1 , if j ≤ p, − , if p < j ≤ p + q, and sign ( σ ( α j )) = +1 , σ ∈ Σ \ { } , j = 1 , . . . , p + q. For σ ∈ Σ let Q σ be the quadratic form on R p + q defined by Q σ ( x ) = p + q X j =1 σ ( α j ) x j . Then Q := Q is a non-degenerate quadratic form of signature ( p, q ) and Q σ , σ = 1, ispositive definite.Let G := SO Q ⊂ GL p + q be the special orthogonal group of Q , i.e., the subgroup of allelements of determinant one leaving Q invariant. Then G is a connected algebraic groupdefined over F . Let J ∈ GL p + q ( R ) be defined by J := diag (cid:0) √ α , . . . , √ α p , p − α p +1 , . . . , p − α p + q (cid:1) . Then the map g J gJ − establishes an isomorphism G ( R ) ∼ = SO( p, q ). Similarly, we have G σ ( R ) ∼ = SO( p + q ), if σ = 1. Let(3.19) G ′ ∼ = Res F/ Q ( G )be the algebraic Q -group obtained by restriction of scalars. There is a canonical isomor-phism of algebraic groups over F (3.20) α : G ′ × Q F ∼ = Y σ ∈ Σ G σ , and the group of real points G ′ ( R ) satisfies G ′ ( R ) ∼ = SO( p, q ) × Y σ ∈ Σ \{ } SO( p + q ) . Denote by(3.21) ∆ : G → Y σ ∈ Σ G σ the diagonal embedding given by ∆( g ) = ( g σ ) σ ∈ Σ .Let O F be the ring of integers of F and let G O F be the group of O F -units of G . Anarithmetic subgroups of G ( F ) is by definition a subgroup which is commensurable with G O F . Let Γ := J G O F J − . Then Γ is a subgroup of SO( p, q ). Lemma 3.1. Γ is a discrete, cocompact subgroup of SO( p, q ) .Proof. For σ ∈ Σ \ { } , the group G σ ( R ) is isomorphic to SO( p + q ). Thus by [BoHa,p. 530], Γ is discrete in SO( p, q ). Since all quadratic forms Q σ , σ = 1, are positivedefinite, the form Q is anisotropic over F . Thus, by [Bo2, page 256] G is anisotropic over F . Therefore, G O F contains no non-trivial unipotent elements. Using [BoHa, Lemma 11.4,Theorem 12.3], it follows that the diagonal image of G O F in Q σ ∈ Σ G σ ( R ) is cocompact andsince G σ ( R ) is compact for σ = 1, Γ is also cocompact in SO( p, q ). (cid:3) Remark . If F = Q [ √ v ] is a real quadratic field, then putting α = · · · = α p = 1, α p +1 = · · · = α p + q = −√ v the above construction has already been given in [Bo1, section4.3].Now we let B be the symmetric bilinear form on F p + q given by B ( e i , e j ) = ( , i + j = p + q + 10 , i + j = p + q + 1 , for e , . . . , e p + q the standard base of F p + q . Let O B be the orthogonal group of B and letSO B be the elements of O B of determinant one. Then O B and SO B are algebraic groupsdefined over F and there exists an isomorphism µ : G ( ¯ F ) → SO B ( ¯ F ), i.e. G is a form ofSO B over F .Let T be the maximal torus of SO B ( ¯ F ) given by T = { diag( λ , . . . , λ n +1 , λ − , . . . , λ − n +1 ) , λ , . . . , λ n +1 ∈ ¯ F ∗ } , where n = ( p + q ) / −
1. Then T is defined over F . Let X ( T ) be the charactergroup of T , written additively. Then a base of X ( T ) is given by the f i : T → ¯ F , f i (diag( λ , . . . , λ n +1 , λ − , . . . , λ − n +1 )) = λ i , where 1 ≤ i ≤ n + 1.By Rep(SO B ( ¯ F )) we denote the finite-dimensional representations of SO B ( ¯ F ) which areirreducible. Then the elements of Rep(SO B ( ¯ F )) correspond bijectively to their highestweights λ τ := m f + · · · + m n +1 f n +1 , where m , . . . , m n +1 ∈ Z , m ≥ m ≥ · · · ≥ m n ≥ | m n +1 | . Since T is split over F , every finite-dimensional irreducible representationof SO B ( ¯ F ) is defined over F , [Ti, Proposition 2.3].For τ ∈ Rep(SO B ( ¯ F )) with highest weight λ τ = m ( τ ) f + · · · + m n +1 ( τ ) f n +1 let τ ′ ∈ Rep(SO B ( ¯ F )) be the element with highest weight λ τ ′ = m ( τ ) f + · · · + m n f n − m n +1 ( τ ) f n +1 .Then the following lemma holds. Lemma 3.3.
For every τ ∈ Rep(SO B ( ¯ F )) there exists a representation ˜ τ of O B ( ¯ F ) thatrestricts to τ + τ ′ on SO B ( ¯ F ) .Proof. The proof of the corresponding proposition for SO B ( C ) given in [GW, Theorem5.22] extends withouth difficulty to any algebraically closed field of characteristic zero. (cid:3) Remark . If τ satisfies τ = τ ′ then there exists in fact a representation of O B thatrestricts to τ . However, we are only interested in the case τ = τ ′ and in this case therepresentation ˜ τ from the previous lemma is irreducible.Now we let PSO B := SO B / {± Id } . Then an element τ ∈ Rep(SO B ( ¯ F )) of highestweight λ τ = m f + · · · + m n +1 f n descends to a representation of PSO B ( ¯ F ) if and only m + · · · + m n +1 is even. Lemma 3.5.
Let τ be a representation of SO B over ¯ F which descends to a representationof PSO B over ¯ F . Then there exists an F -rational representation of G which over ¯ F isequivalent to ( τ + τ ′ ) ◦ µ .Proof. For σ ∈ Gal( ¯
F /F ) define an automorphism φ σ of SO B ( ¯ F ) by φ σ := µ ◦ σ ◦ µ − ◦ σ − . Since the Dynkin diagram D n +1 has exactly one non-trivial automorphism, thereis a natural isomorphism Aut(SO B ( ¯ F )) ∼ = PO B ( ¯ F ), where PO B ( ¯ F ) acts on SO B ( ¯ F ) byconjugation, and thus there exists a unique a σ ∈ PO B ( ¯ F ) such that for each g ∈ SO B ( ¯ F )one has φ σ ( g ) = a σ ga − σ . Thus one has µ ( σ ( g )) = a σ σ ( µ ( g )) a − σ . (3.22)We can regard the assignment σ → a σ as an element of the first Galois-cohomology set H (Gal( ¯ F /F ) , PO B ( ¯ F )). By Lemma 3.3 there exist a representation ˜ τ of PO B ( ¯ F ) on V ˜ τ = V τ ⊕ V τ ′ which restricts to τ ⊕ τ ′ on SO B ( ¯ F ). The assignment σ ˜ τ ( a σ ) is anelement of H (Gal( ¯ F /F ) , GL( V ˜ τ )) and since this set is trivial by Hilbert’s theorem 90,there exists an x ∈ GL( V ˜ τ ) such that˜ τ ( a σ ) = x − σ ( x ) ∀ σ ∈ Gal( ¯
F /F ) . (3.23)Now define a representation ρ of G ( ¯ F ) by ρ := Int( x ) ◦ ( τ + τ ′ ) ◦ µ . Then ρ is equivalentto ( τ + τ ′ ) ◦ µ . Applying (3.22) and (3.23) it follows that for σ ∈ Gal( ¯
F /F ) and g ∈ G ( ¯ F )one has ρ ( σ ( g )) = x ˜ τ ( a σ )( τ + τ ′ )( σ ( µ ( g )))˜ τ ( a − σ ) x − = σ ( x ) σ (( τ + τ ′ )( µ ( g ))) σ ( x ) − = σ ( ρ ( g )) , where we used that τ + τ ′ is defined over F and hence commutes with Gal( ¯ F /F ). Thus ρ commutes with Gal( ¯ F /F ) and thus it is defined over F . (cid:3) Now we may fix an embedding of SO ( p, q ) into SO B ( C ) such that the representationsof SO ( p, q ) with highest weight m e + · · · + m n +1 e n +1 are the restrictions to SO ( p, q ) ofthe representation of SO B ( C ) with highest weight m f + · · · + m n +1 f n +1 .The following proposition is certainly well known and was used already by Bergeronand Venkatesh [BV, section 8.1]. However, for the sake of completeness we include aproof here. If V is a finite-dimensional F -vector space, let V σ be the F -vector space withscalar-multiplication a · v := σ ( a ) v , a ∈ F , v ∈ V . Lemma 3.6.
Let G ′ be an algebraic group defined over Q . Let V be a finite-dimensional F -vector space and let ρ : G ′ → GL( V ) be a representation defined over F . Then ˜ ρ := Q σ ∈ Σ ρ σ − is defined over Q , where ρ σ − is regarded as an F -rational representation of G ′ on V σ .Proof. Each σ ∈ Σ acts on Q σ ∈ Σ V σ as a σ -linear automorphism by permuting the factors.The corresponding Q -structure of Q σ ∈ Σ V σ is V , regarded as a Q -vector space and embed-ded diagonally into Q σ ∈ Σ V σ . Now it is easy to see that ˜ ρ commutes with the action ofΣ on G ′ and the action of Σ on GL( Q σ ∈ Σ V σ ) associated to this Q -structure. Thus ˜ ρ isdefined over Q . (cid:3) Let G ′ ( R ) be the connected component of 1 ∈ G ′ ( R ) and G ′ ( R ) c := Q σ ∈ Σ \{ } SO( p + q ).Then we have G ′ ( R ) ∼ = SO ( p, q ) × G ′ ( R ) c . Let θ be the standard Cartan-involution of SO ( p, q ). Then θ ⊗ Id G ′ ( R ) c is a Cartan in-volution of G ′ ( R ) which we continue to denote by θ . By Rep( G ′ ( R ) ) we denote thefinite-dimensional irreducible complex representations of G ′ ( R ) . For τ ∈ Rep( G ′ ( R ) ), let τ θ be the element of Rep( G ′ ( R ) ) defined by τ θ := τ ◦ θ . Proposition 3.7.
There exists a sequence ρ ( m ) of Q -rational representations of G ′ onfinite-dimensional Q -vector spaces V ρ ( m ) such that(1) For the decomposition (3.24) ρ ( m ) = M τ ∈ Rep( G ′ ( R ) ) [ ρ ( m ) : τ ] τ, [ ρ ( m ) : τ ] ∈ N of ρ ( m ) , regarded as a complex representation of G ′ ( R ) on thevector space V ρ ( m ) ⊗ Q C , into irreducible representations of G ′ ( R ) one has τ = τ θ for each τ ∈ Rep( G ′ ( R ) ) with [ ρ ( m ) : τ ] = 0 .(2) The dimension dim( V ρ ( m ) ) is a polynomial in m with leading term dim( V ρ ( m ) ) = C d m dn ( n +1) / + O ( m d ( n +1) / − ) , as m → ∞ , where C > is a constanst which depends only on n .Proof. The Galois group Σ acts on Q σ ∈ Σ G σ as follows. For g ∈ Q σ ∈ Σ G σ and σ ∈ Σ wedenote the projection of g to G σ by g σ . Then for σ, σ ′ ∈ Σ one has( σg ) σ ′ = σ ( g σ − σ ′ ) . Now assume that for each σ ∈ Σ we are given a finite-dimensional F -vector space V ρ ( σ ) and a representation ρ ( σ ) of G σ on V ρ ( σ ) , defined over F . Then the tensor-product ρ := O σ ∈ Σ ρ ( σ )is a representation of Q σ ∈ Σ G σ on N σ ∈ Σ V ρ ( σ ) and it follows that for σ ′ ∈ Σ one has ρ σ ′ = O σ ∈ Σ ρ ( σ ′− σ ) σ ′ . (3.25) Now if n is even, for m ∈ N we let τ ( m ) be the representation of G over ¯ F of highestweight 2 me + · · · + 2 me n +1 . If n is odd, we let τ ( m ) be the representation of highestweight me + · · · + me n +1 . Then τ ( m ) and τ ( m ) θ descend to representations of P G . Thusby Lemma 3.5 there exists a representation of G over F which over ¯ F is equivalent to τ ( m ) + τ ( m ) θ . Thus if we define ρ ( m ) by ρ ( m ) := O σ ∈ Σ ( τ ( m ) + τ ( m ) θ ) σ , (3.26)then ρ ( m ) is defined over F and by (3.25), ρ ( m ) is equivalent to ρ ( m ) σ for each σ ∈ Σ.Hence if we let ρ ( m ) be the direct sum of d copies of ρ ( m ) then by Lemma 3.6 ρ ( m )is defined over Q . Each irreducible component of ρ ( m ) | G ′ ( R ) , regarded as a complexrepresentation of G ′ ( R ) on V ρ ( m ) ⊗ Q C , is of the form τ ( m ) ⊗ π , or τ ( m ) θ ⊗ π ′ , where π and π ′ are irreducible representations of G ′ ( R ) c . Since τ ( m ) and τ ( m ) θ are not θ -invariant, thesame holds for each irreducible component of ρ ( m ) | G ′ ( R ) . This proves the first statement.The second statement follows from Weyl’s dimension formula. (cid:3) We can now turn to the proof of Theorem 1.1. Let ∆ be the diagonal embedding of G into Q σ ∈ Σ G σ . Then we can choose the isomorphism α in (3.20) such that α ( G ′ Z ) = ∆( G O F ). LetΓ ⊂ G O F be a subgroup of finite index. Via the isomorphism G ( R ) ∼ = SO( p, q ) we identifyΓ with a subgroup of SO( p, q ). We choose Γ such that it is torsion free and is contained inSO ( p, q ). By Lemma 3.1, Γ is a cocompact lattice in SO ( p, q ). Let Γ ′ = ∆(Γ). Since Γand Γ ′ are isomorphic, it suffices to prove the statements of Theorem 1.1 for Γ ′ .The group K := SO( p ) × SO( q ) is a a maximal compact subgroup of SO ( p, q ). Put K ′ := K × Y σ ∈ Σ σ =1 SO( p + q ) . Then K ′ is a maximal compact subgroup of G ′ ( R ) . Put ˜ X ′ := G ′ ( R ) /K ′ and X ′ := Γ ′ \ ˜ X ′ .Let ( ρ ( m ) , V ρ ( m ) ) be the sequence of Q -rational representations of G ′ of Proposition 3.7.Since each ρ ( m ) is defined over Q , there exists a free Z -module M ρ ( m ) in V ρ ( m ) which isstable under ρ ( m )(Γ ′ ) and such that M ρ ( m ) ⊗ Z Q ∼ = V ρ ( m ) , see for example [PR, page 173].Let V C ρ ( m ) := V ρ ( m ) ⊗ Q C . Then the restriction of ρ ( m ) to Γ ′ induces the flat complex vectorbundle E ρ ( m ) := ˜ X ′ × ρ ( m ) | Γ ′ V C ρ ( m ) over X ′ . The decomoposition (3.24) of ρ ( m ) induces a corresponding decomposition of E ρ ( m ) into the direct sum of complex vector bundles associated to the restriction to Γ ′ ofirreducible finite-dimensional representations τ of G ′ ( R ) . By Proposition 3.7, each τ with[ ρ ( m ) : τ ] = 0 satisfies τ = τ θ and thus by [BW, Chapter VII, Theorem 6.7] one has H ∗ ( X ′ ; E ρ ( m ) ) = 0 , (3.27)where H ∗ ( X ′ ; E ρ ( m ) ) denotes the de Rham cohomology with coefficients in E ρ ( m ) . Chosea Hermitian fibre metric in E ρ ( m ) . Let T X ′ ( ρ ( m )) be the analytic torsion of X ′ and ρ ( m ) (see (1.9)). It follows from (3.27) that T X ′ ( ρ ( m )) is metric independent [M¨u1, Corollary2.7]. Moreover H ∗ (Γ ′ , M ρ ( m ) ) is a finite abelian group and by (1.10) we have(3.28) log T X ′ ( ρ ( m )) = n X q =0 ( − q +1 log | H q (Γ ′ , M ρ ( m ) ) | . This equality reduces the proof of Theorem 1.1 to the study of the asymptotic behaviorof T X ′ ( ρ ( m )) as m → ∞ , which is exactly the problem that has been dealt with in [MP].Since { ρ ( m ) } is not a sequence of representations that has been considered in [MP], wecannot apply the results of [MP] directly. We first need to reduce it to a case to which[MP] can be applied.Let ρ ( m ) be defined by (3.26) and let T X ′ ( ρ ( m )) be the corresponding analytic torsion.Since ρ ( m ) is the direct sum of d copies of ρ ( m ), we getlog T X ′ ( ρ ( m )) = d log T X ′ ( ρ ( m ))Now let T (2) X ′ ( ρ ( m )) be the L -torsion with respect to ρ ( m ) (see [MP, section 5]). If weapply [MP, Proposition 1.2] to the irreducible components of ρ ( m ), it follows that thereexists c > T X ′ ( ρ ( m )) = log T (2) X ′ ( ρ ( m )) + O ( e − cm ) , as m → ∞ . Using the definition of ρ ( m ) by (3.26), [MP, (5.21)] and [MP, Proposition5.3], it follows thatlog T (2) X ′ ( ρ ( m )) = (cid:16) log T (2) X ′ ( τ ( m )) + log T (2) X ′ τ ( m ) θ (cid:17) (2 dim τ ( m )) d − . If C p,q is as in (1.3), then by [MP, Proposition 6.7] one haslog T (2) X ′ ( τ ( m )) = log T (2) X ′ ( τ ( m ) θ ) = C p,q vol( X ′ ) m dim τ ( m ) + O (dim( τ ( m )) , as m → ∞ . Thus putting everything together we obtainlog T X ′ ( ρ ( m )) = C p,q vol( X ′ ) m dim( ρ ( m )) + O (dim( ρ ( m ))) , as m → ∞ . Since X ∼ = X ′ and H ∗ (Γ , M ρ ( m ) ) ∼ = H ∗ (Γ ′ , M ρ ( m ) ), Theorem 1.1 follows from(3.28) and the second statement of Proposition 3.7.4. Arithmetic subgroups of SL ( R )Let D be a nine-dimensional division algebra over Q . Then by the Brauer-Hasse-Noethertheorem [Ro], D is a cyclic algebra for a cubic extension L of Q . Moreover, L splits D , i.e.there exists an isomorphism of L -algebras φ : D ⊗ Q L ∼ = Mat × ( L ) . (4.29)Thus for x ∈ D the reduced norm N ( x ) is given by N ( x ) := det( φ ( x ⊗ G := SL ( D ), where SL ( D ) := { x ∈ D : N ( x ) = 1 } . Then by [PR, 2.3.1], G has a canonical structure of an algebraic group defined over Q .We regard SL as an algebraic group over Q . The isomorphism φ from (4.29) induces anisomorphism φ : G ( L ) ∼ = SL ( L ) , i.e. G is a form of SL over L . Moreover, the following Lemma holds. Lemma 4.1.
Let ρ be a Q -rational representation of SL . Then there exists a Q -rationalrepresentation of G which over L is equivalent to ρ ◦ φ .Proof. By [PR, Proposition 2.17], G is an inner form of SL . Thus the proof of Lemma 3.1in [MaM¨u] can be generalized without difficulties to prove the Lemma. (cid:3) Let o be an order in D , i.e., o is a free Z -submodule of D which is generated by a Z -baseof D and which is also a subring of D . Put o := { x ∈ o : N ( x ) = 1 } . The left regular representation of D on itself induces a Q -rational representation of G on D , see [PR, 2.3.1] and the stabilizer of o is o . Thus o is arithmetic subgroup of G ( Q ).Put Γ := φ ( o ) . Then Γ is an arithmetic subgroup of SL ( R ). Moreover, the following lemma holds. Lemma 4.2.
The group Γ is a cocompact subgroup of SL ( R ) .Proof. By [PR, Proposition 2.12], G is anisotropic over Q . Thus the proposition followsfrom [BoHa, Lemma 11.4, Theorem 11.8]. (cid:3) Let T be the standard maximal torus in SL consisting of the diagonal matrices ofdeterminant 1. Then T is defined over Q and is Q -split. Let t be the Lie-algebra of T ( R )consisting of all diagonal matrices of trace 0. Let e i ∈ t ∗ be defined by e i (diag( t , t , t )) := P j =1 δ i,j t j . Then with respect to the standard odering of the roots of t C in sl , C thefundamental weights ω , ω ∈ ˜ t ∗ C are given by ω = 23 ( e − e ) + 13 ( e − e ); ω = 13 ( e − e ) + 23 ( e − e ) . (4.30)The finite-dimensional irreducible representations τ of SL ( R ) are parametrized by theirhighest weights Λ τ = m ω + m ω . If θ is the standard Cartan involution of SL ( R ), thenthe highest weight of the representation τ θ := τ ◦ θ is given by Λ τ θ = m ω + m ω . Proposition 4.3.
Let τ be a finite-dimensional irreducible representation of SL ( R ) on afinite-dimensional vector space V τ . Then there exists a lattice M in V τ which is invariantunder τ (Γ) . Proof.
Since T is Q -split τ is defined over Q [Ti][Proposition 2.3]. By Lemma 4.1, thereexists a rational representation τ ′ of G on a finite-dimensional Q -vector space V ( τ ′ ) whichover L is equivalent to τ ◦ φ . Since o is an arithmetic subgroup of G ( Q ), there existsa lattice in V ( τ ′ ) which is stable under τ ′ ( o ), see for example [PR, page 173]. SinceΓ = φ ( o ), the Proposition follows. (cid:3) We can now turn to the proof of Theorem 1.3. Let e X = SL ( R ) / SO(3) and X = Γ \ e X ,where Γ ⊂ SL ( R ) is an arithmetic subgroup as above. Chose a SL ( R )-invariant metricon e X and equip X with the induced metric. Let Λ ∈ h ∗ C be a highest weight. Assume thatΛ satisfies Λ = Λ θ . Then the same holds for each weight m Λ, m ∈ N . Let τ Λ ( m ) be theirreducible finite-dimensional representation on V Λ ( m ) with highest weight m Λ. Let E τ Λ ( m ) be the flat vector bundle over X associated to τ Λ ( m ). By [BW, Chapter VII, Theorem 6.7]we have(4.31) H ∗ ( X, E τ Λ ( m ) ) = 0 . Let T X ( τ λ ( m )) be the analytic torsion with respect to any Hermitian fibre metric in E τ Λ ( m ) .By (4.31) and [M¨u1, Corollary 2.7], T X ( τ Λ ( m )) is independent of the choice of metrics on X and in E τ Λ ( m ) . Let M Λ ( m ) ⊂ V Λ ( m ) be an arithmetic Γ-module, which exists byProposition 4.3. By (4.31), H ∗ ( X, M Λ ( m )) is a finite abelian group and by (1.10) we have(4.32) log T X ( τ λ ( m )) = X q =0 ( − q +1 log | H q ( X, M Λ ( m )) | . Using Theorem 1.1 and Corollary 1.5 of [MP], the proof of Theorem 1.3 follows.
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