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DIVISION ALGEBRAS AND NON-COMMENSURABLE ISOSPECTRAL MANIFOLDS 5 0 ALEXANDER LUBOTZKY, BETH SAMUELS, AND UZI VISHNE 0 2 n a J Abstract. A. Reid [R] showed that if Γ1 and Γ2 are arithmetic 5 latticesinG=PGL2(R)orinPGL2(C)whichgiverisetoisospec- ] tral manifolds, then Γ1 and Γ2 are commensurable (after conju- P gation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S . S = PGLd(C)/PUd(C), the situation is quite different: there are h arbitrarily large finite families of isospectral non-commensurable t a compact manifolds covered by S. m The constructions are based on the arithmetic groups obtained [ from division algebras with the same ramification points but dif- 1 ferent invariants. v 4 6 0 1. Introduction 1 0 Let X be a compact Riemannian manifold and ∆ = ∆ its Lapla- 5 X 0 cian. The spectrum of X, spec(X), is the multiset of eigenvalues of ∆ / h acting on L2(X). This is a discrete subset of R. If Y is another such t a object, we say that X and Y are isospectral (or, sometimes, cospec- m tral) if spec(X) = spec(Y). The problem of finding isospectral non- : v isomorphicobjects has along history, startingwith MarcKac’sseminal i X paper [Ka], “Can one hear the shape of a drum?” (see [Go] and the r a references therein). The most powerful and general method of obtaining such pairs is due to Sunada [Su]: he showed that if X covers X with a finite Galois 0 group H, i.e. X = X /H, and if H and H are subgroups of H such 0 1 2 that for every conjugacy class C ⊂ H, |C ∩H | = |C ∩H |, then 1 2 X = X /H and X = X /H are isospectral. Of course, one still has 1 0 1 2 0 2 to determine whether X and X are isomorphic (for example, X and 1 2 1 X will be isomorphic if H is conjugate to H , but not only in such a 2 1 2 Date: Submitted: Dec. 30, 2004. 1 2 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE case). Sunada’s method has been implemented in many situations (e.g. [Br1], [Br2]), and, in particular, has led to the solution of the original problem of M. Kac (see [GWW]). A common feature to all the isospectral pairs, X and X , con- 1 2 structed by Sunada’s method, is that they are commensurable, that is, X and X have a common finite cover X . So, while X and X 1 2 0 1 2 are not isomorphic, they are ‘virtually isomorphic’. Sunada implemented his method to obtain non-isomorphic isospectral Riemann surfaces. He gave examples of torsion free cocompact lattices Γ(cid:1)Γ in G = PGL (R) with Γ /Γ ∼= H and H = Γ /Γ, and 0 2 0 1 1 H = Γ /Γ, as above. His method implies that the G-representa- 2 2 tions L2(Γ \G) and L2(Γ \G) are isomorphic, which demonstrates that 1 2 Γ \H2 isisospectraltoΓ \H2. HereH2 = PGL (R)/K istheassociated 1 2 2 symmetric space, and K is the maximal compact subgroup PO (R) of 2 G. Vigneras was the first to present examples of isospectral Riemann surfaces in [V], where she used the theory of quaternion algebras. Her examples are also commensurable to each other. In fact, it is still an open problem whether every two isospectral Riemann surfaces are commensurable, or equivalently whether every two cocompact lattices Γ and Γ in G = PGL (R), such that Γ \G/K and Γ \G/K are 1 2 2 1 2 isospectral, are commensurable (after conjugation, i.e. for some g ∈ G, [Γ :g−1Γ g ∩Γ ] < ∞). But, Alan Reid [R] showed that if Γ and Γ 2 1 2 1 2 are such arithmetic lattices, then indeed Γ and Γ are commensurable. 1 2 He also showed a similar result for G = PGL (C). 2 Our main result shows that the situation is quite different for G = PGL (R) or PGL (C) if d ≥ 3. d d Theorem 1. Let F = R or C, G = PGL (F) where d ≥ 3, K ≤ G d a maximal compact subgroup, and S = G/K the associated symmetric space. Then for every m ∈ N, there exists a family of m torsion free cocompact arithmetic lattices {Γ } in G, such that Γ \S are i i=1,...,m i isospectral and not commensurable. Taking K = PO (R) in the real case and K = PU (C) in the com- d d plex case, the quotients Γ \G/K are isospectral compact manifolds i DIVISION ALGEBRAS AND ISOSPECTRALITY 3 covered by the symmetric space S = PGL (F)/K which is equal to d SL (R)/SO (R) or SL (C)/SU (C), respectively. The covering map d d d d S→Γ \S is a local isomorphism, since the Γ are torsion free. i i This result should be compared with Spatzier’s construction [Sp] of isospectrallocallysymmetric spaces ofhighrank. However, hisisospec- tral examples are always commensurable to each other. Let us now outline the method of proof. The following is well known (see for example [Pe]). Proposition 2. Let G be a semisimple group, K ≤ G a maximal compact subgroup, and Γ ,Γ ≤ G discrete cocompact subgroups. If 1 2 (1) L2(Γ \G) ∼= L2(Γ \G) 1 2 as (right) G-representations, then Γ \G/K and Γ \G/K are isospectral 1 2 quotients of G/K (indeed, they are even strongly isospectral, i.e. with respect to the higher dimensional Laplacians). In some special cases, the inverse of Proposition 2 is known to be true (cf. [Pe] and the references therein, and [D]). To prove Theorem 1 for G = PGL (F) (F = R or C), we choose d the discrete subgroups Γ to be arithmetic lattices of inner forms, as i follows. Let k be a global field and {k } the completions with respect ν to its valuations. Recall that Br(k ) ∼= Q/Z for a non-archimedean ν valuation, while Br(R) ∼= 1Z/Z and Br(C) = 0, where Br() denotes 2 the Brauer group of a field. For a division algebra D, [D] ∈ Br(k), the value associated to D⊗ k is called the ‘local invariant’ at ν. By k ν Albert-Brauer-Hasse-Noether theorem, [D] 7→ ([D⊗ k ]) defines an k ν ν injective map for the Brauer groups (2) Br(k)−→⊕Br(k ), ν and the image of this map is composed of vectors whose sum is zero. Over local (and thus also global) fields, the degree of a division algebra is equal to its exponent (which is the order of its equivalence class in the Brauer group). Fix d > 1 and let T = {θ ,...,θ } be a finite non-empty set of 1 t non-archimedean valuations of k. Let a ,...,a ,b ,...,b ∈ N be such 1 t 1 t 4 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE that for every j = 1,...,t, 0 ≤ a ,b < d, (a ,d) = (b ,d) = 1, and j j j j a ≡ b ≡ 0 (mod d). Let D (resp. D ) be the unique division P j P j 1 2 algebra of degree d (i.e. dimension d2) over k which ramifies exactly at T,andwhoseinvariantatk isa /d(resp. b /d). ThusD andD split θj j j 1 2 at every ν 6∈ T. Let G′ be the algebraic group D×/Z× (i = 1,2) where i i Z is the center, and G = PGL . Since G′(k ) splits for every ν 6∈ T, d i ν the groups G′(A ) and G′(A ) are equal where A = ′ k . We 1 Tc 2 Tc Tc Qν6∈T ν identify these groups with G = G(A ). Here, A is the ring of adèles 0 Tc and ′ denotes the restricted product, namely the vectors (x ) such Q ν that ν(x ) ≥ 0 for almost every ν. ν A basic result is Theorem 3. Let G = PGL (A ) be the group defined above, and let 0 d Tc ∆ = G′(k) (i = 1,2), cocompact lattices in G . The spaces L2(∆ \G ) i i 0 1 0 and L2(∆ \G ) are isomorphic as G -representations. 2 0 0 The result follows by comparing trace formulas. This was done ex- plicitly for the case where d is prime in [GJ, Theorem 1.12] and in [Bu, Theorem 54]. It seems that the general case is also known to experts, but we were unable to locate a reference. For the sake of complete- ness we show in Section 4 how the result can be deduced (for arbitrary d) from the global Jacquet-Langlands correspondence, as described in [HT]. (The proof there is given for the characteristic zero case, and we refer the reader to [LSV1, Remark 1.6] for some details about the positive characteristic case). We should stress that in the current paper, we are using only the characteristic zero case. We prefer to give here the more general for- mulation, preparing for a subsequent paper [LSV2], in which we use analogous ideas over a local field of positive characteristic to construct isospectral simplicial complexes and isospectral Cayley graphs of some finite simple groups. One deduces the isospectrality in Theorem 1 by applying strong approximation to Theorem 3. This is done in Section 2. To prove the non-commensurability, we first show (Theorem 9) that two division k- algebras D and D give rise to commensurable arithmetic lattices in 1 2 PGL (F) iff D (k) and D (k) are isomorphic or anti-isomorphic (as d 1 2 DIVISION ALGEBRAS AND ISOSPECTRALITY 5 rings, rather than as k-algebras). We then analyze when two such division algebras are isomorphic (or anti-isomorphic) as rings, and show that we can produce as many examples as we need of non-isomorphic division rings. This is done in Section 3. Finally, we mention that our work is very much in the spirit of the paper of Vigneras [V], who used a similar idea to find isospectral Rie- mann surfaces. However, as mentioned above, her examples, which are lattices in PGL (R), are commensurable, as they should be by Reid’s 2 theorem. The case d = 2 differs from those of d ≥ 3 in that a global division algebra of degree 2 is determined by its ramification points, while for d ≥ 3 there are non-isomorphic division algebras with the same ramification points. We thank G. Margulis for helpful discussions, A.S. Rapinchuk for his helpwithTheorem9,andD.GoldsteinandR.Guralnickforsimplifying theproofofLemma10. Theauthorsalsoacknowledgesupportofgrants by the NSF and the U.S.-Israel Binational Science Foundation. 2. Isospectrality Although for the proof of Theorem 1 we may assume characteristic zero, we are stating and proving the results of this section for positive characteristic as well. Therefore, we obtain Theorem 1 together with: Theorem 4. Let F be a local field of positive characteristic, G = PGL (F) where d ≥ 3, K a maximal compact subgroup and B (F) = d d G/K the associated Bruhat-Tits building. Then for every m ∈ N there exists a family of m torsion free cocompact arithmetic lattices {Γ } in G, such that the finite complexes Γ \B (F) are isospec- i i=1,...,m i d tral and not commensurable. Notice that Theorems 1 and4 cover the archimedean localfields, and the non-archimedean local fields of positive characteristic. Our meth- ods do not apply to non-archimedean local fields F of zero characteristic, where there are no cocompact lattices of inner type in PGL (F) d (d ≥ 3). There are cocompact lattices of outer type in these groups; however we leave open the following problem: 6 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE Question 5. Let F be a non-archimedean local field of zero characteristic, G = PGL (F) where d ≥ 3, K a maximal compact subgroup d and B (F) = G/K the associated Bruhat-Tits building. Do there exist d torsion free cocompact arithmetic lattices {Γ } in G, such that the i i=1,2 finite complexes Γ \B (F) are isospectral and not commensurable? i d Let k be a global field, and assume it has at most one archimedean valuation. Let ν denote the archimedean valuation if chark = 0, and 0 a valuation of degree 1 if k is a function field (for k = F (t), ν is q 0 either the minus-degree valuation or is determined by a linear prime). Let F = k . (For F = R or F = C, we take k = Q or k = Q[i], ν0 respectively). Let V denote the set of all valuations of k other than ν . For a 0 valuation ν ∈ V, O is the ring of integers in the completion k , and ν ν P is the valuation ideal. ν Let T ⊆ V be a finite subset, and define D and G′ as in the intro- i i duction (i = 1,2). Let G denote the algebraic group PGL . Notice d that ν 6∈ T, so in particular G′(F) = G(F) (i = 1,2). Let A denote 0 i T the direct product k , so that A = A ×A is the ring of adèles Qθ∈T θ T Tc over k. Let R = {x ∈ k: ν(x) ≥ 0 for every ν 6= ν }. (In the cases k = Q 0 0 or k = Q[i], with ν as the archimedean valuation, we obtain R = Z 0 0 or R = Z[i], respectively.) Then, for every θ ∈ T, we can choose 0 a uniformizer ̟ such that ν(̟ ) = 0 for every ν 6∈ {ν ,θ} (so in θ θ 0 particular, ̟ ∈ R ). In this situation, θ 0 (3) R = {x ∈ k: ν(x) ≥ 0 for every ν 6∈ T ∪{ν }} 0,T 0 is equal to R [{̟−1} ]. Fix i ∈ {1,2}. Since division algebras over 0 θ θ∈T global fields are cyclic, there is a Galois field extensions k /k of dimen- i sion d contained in D (k). If φ ∈ Gal(k /k) is a generator, there is an i i element b ∈ k× such that D = k [z| zaz−1 = φ(a), zd = b ]. Taking i i i i an integral basis, one finds an order O ⊆ k which is a cyclic extension i i (of rings) of R . The constant b can be chosen to be in the multi- 0,T i plicative group generated by the uniformizers ̟ . Thus, b is invertible θ i in R . So O [z] (with the relations as above) is an Azumaya algebra 0,T i of rank d over the center R , and D = k⊗ O [z]. The group of 0,T i R0,T i DIVISION ALGEBRAS AND ISOSPECTRALITY 7 invertible elements in O [z] (again with the above relations), modulo i its center, is G′(R ). i 0,T If an algebraic group G′ is defined over a ring R and 0 6= I(cid:1)R is an ideal, we have the principal congruence subgroups (4) G′(R,I) = Ker(G′(R)→G′(R/I)). The congruence subgroups G′(R ,I) are discrete in G(F), and co- i 0,T compact if T 6= ∅ [PlR]. Let 0 6= I(cid:1)R . Define a function r:V−(T ∪ {ν })→N ∪ {0} by 0,T 0 setting (5) r = min{ν(a): a ∈ I}, ν and let (6) U(r) = Y G(Oν,Pνrν), ν∈V−(T∪{ν0}) an open compact subgroup of G(A ). For almost every ν, r = 0, Tc−{ν0} ν and then P0 = O and G(O ,Prν) = G(O ) by definition. Let ν ν ν ν ν (7) U = U(r)G′(A ), i i T an open compact subgroup of G′(A). i Lemma 6. The quotient G′(A)/(G′(k)G(F)U ) is a d-torsion finite i i i abelian group, which is independent of i. Proof. Recallthe reduced normmapD (k)×→k×, which coincides with i the determinant over splitting fields of k. Let G denote the simply i connected cover of G′, i.e. G is the subgroup of elements of reduced i i norm 1 in D×. For every field k ⊇ k, the image of the covering map i 1 Ψ:G (k )→G′(k ) is co-abelian in G′(k ). The strong approximation i 1 i 1 i 1 theorem [PlR] (see also [LSV1, Subsection 3.2]) applies to G , and so i G (k)G (F)U = G (A) where i i i U = Y Gi(Oν,Pνrν)×Gi(AT), ν∈V−(T∪{ν0}) and G (O )U is open compact in G (A). i ν0 i The reduced norms G′(k )→k×/k×d is onto whenever G′ splits at i ν ν ν i k (as G′(k ) = PGL (k )), and also whenever k is non-archimedean ν i ν d ν ν 8 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE (since over local fields the norm map in a division algebra is onto, [Pi, Chapter 17]). In the second case, the restriction to G′(O )→O×/O×d i ν ν ν is also onto. Thus, the norm map G′(A)→A×/A×d is onto, and induces i an isomorphism Φ:G′(A)/Ψ(G (A))→A×/A×d. i i Since Ψ maps G (k), G (F) and U into G′(k), G(F) and U , respec- i i i i tively, we have Ψ(G (A)) = Ψ(G (k)G (F)U) ⊆ G′(k)G(F)U . We i i i i i need to compute the image of G′(k)G(F)U under Φ. For a valua- i i tion ring O with maximal ideal P and r ≥ 0, let O(r) denote the subgroup {a ∈ O×: a ≡ 1 (mod Pr)}. Now, modulo d powers, the determinant maps G(F) to F× and G(O ,Prν) to O(rν). It follows ν ν ν that Φ takes G′(k) to k×A×d, G(F) to F×A×d, and U to the product i i ( O(rν) × k×)A×d. We can now compute that Qν∈V−T ν Qθ∈T θ G′i(A)/(G′i(k)G(F)Ui) ∼= A×/(k×F×A×d Y Oν(rν)Ykθ×) ν∈V−(T∪{ν0}) θ∈T ∼= A× /(k×F×A× d Y O(rν)). Tc Tc ν ν∈V−(T∪{ν0}) Recall that k× ∼= Z×O×, where the Z summand is the value group. ν ν Dividing the numerator and denominator in the last expression by O×d, the quotient is Qν∈V−(T∪{ν0}) ν ′ Yk×/(k×Y′O(rν)k×d) ∼= ν ν ν ′ ′ ∼= Y(Z×O×)/(k×Y(dZ×O(rν)O×)) ν ν ν ′ ′ ∼= Y(Z×O×/O×d)/(k×Y(dZ×O(rν)O×/O×d)) ν ν ν ν ν ′ ′ ∼= (YZYO×/O×d)/(k×YdZYO(rν)O×/O×d) ν ν ν ν ν ′ ∼= (Y(Z/d×O×/O(rν)O×d))/k×, ν ν ν where all products are over ν ∈ V − (T ∪ {ν }) and ′ denotes the 0 Q restricted product (such that ν(a ) ≥ 0 almost always). The quo- ν tients O×/O(rν) are always finite, and r = 0 for almost all ν, hence ν ν ν DIVISION ALGEBRAS AND ISOSPECTRALITY 9 O×/O(rν)O×d = 0 for all but finitely many valuation. In the last ex- ν ν ν pression,k× embedsintheν componentbya 7→ (ν(a),a̟ −ν(a)O(rν)O×d), ν ν ν and in particular the components with O×/O(rν)O×d = 0 vanish in the ν ν ν quotient. Therefore we have a finite product of finite groups, and the (cid:3) result follows. Proposition 7. Let 0 6= I(cid:1)R . There is an isomorphism 0,T (8) L2(G′(R ,I)\G(F)) ∼= L2(G′(R ,I)\G(F)), 1 0,T 2 0,T as representations of G(F) = PGL (F). d Proof. Weprove thisstatement bytransferring it to itsadèlicanalogue. Let U be the group defined in Equation (7). Then G(O )U is open i ν0 i compact in G′(A), and i (9) G′(R ,I) = G′(k)∩G(F)U , i 0,T i i the intersection taken in G′(A) and then projected to the G(F) com- i ponent. Therefore G′(R ,I)\G(F) is related to G′(k)\G′(A)/U . i 0,T i i i Since G = G(A ) by definition, G′(A)/U = G′(A )/U(r) = 0 Tc i i i Tc G /U(r), and 0 G′(k)\G′(A)/U ∼= G′(k)\G /U(r). i i i i 0 Consequently, L2(G′(k)\G )U(r) ∼= L2(G′(k)\G /U(r)) ∼= L2(G′(k)\G′(A)/U ). i 0 i 0 i i i Now let H = G′(k)G(F)U . From Equation (9) it follows that i i i G′(R ,I)\G(F) ∼= G′(k)\H /U . i 0,T i i i By the lemma, C = G′(A)/H is finite (d-torsion) abelian, which is i i independent of i. Since G(F) commutes with U and is contained in i H , we have the decomposition i L2(G′(k)\G′(A)/U ) ∼= ⊕ L2(G′(k)\H /U ) i i i |C| i i i as G(F)-representations. Thus L2(G′(k)\G )U(r) ∼= ⊕ L2(G′(k)\H /U ) ∼= ⊕ L2(G′(R ,I)\G(F)). i 0 |C| i i i |C| i 0,T ByTheorem3,L2(G′(k)\G ) ∼= L2(G′(k)\G )asrepresentationspaces, 1 0 2 0 (cid:3) so the result follows. 10 ALEXANDERLUBOTZKY,BETH SAMUELS, ANDUZI VISHNE Now, let 0 6= I(cid:1)R , and take Γ = G′(R ,I). By Proposition 0,T i i 0,T 2 and Proposition 7, we see that Γ \G(F)/K and Γ \G(F)/K are 1 2 isospectral, where K is a fixed maximal compact subgroup of G(F). If I issmall enough, then Γ and Γ aretorsion free, andso the projection 1 2 G(F)/K→Γ \G(F)/K is a local isomorphism. i Remark 8. When T is fixed, the choice of U(r) determines the family of quotients. When U(r)′ ⊆ U(r), the quotients corresponding to U(r) are covered by those corresponding to U(r)′. In this sense, we find not only families of isospectral structures, but infinite “inverse limits” of such families. 3. Non-commensurability As in the previous section, we state and prove the results of this section for arbitrary characteristic, proving Theorems 1 and 4 together. To show that Γ \G(F)/K are not commensurable, we need the fol- i lowing theorem. Recall that the maximal compact subgroup K of PGL (F) is taken to be K = PO (R) if k = R, K = PU (C) if k = C, d d d and K = PGL (O) (where O is the ring of integers of F = k ) if d ν0 chark > 0. Let D and D be two central division algebras over k, as 1 2 in the previous section, and G′ the corresponding algebraic groups. Let i T and T be the ramification points of D and D (this time not nec- 1 2 1 2 essarily equal), and let R be the subrings of k defined in Equation 0,Ti (3) (for T rather than T). Note that if σ:D (k)→D (k) is an isomor- i 1 2 phism or anti-isomorphism of rings, then σ induces an automorphism of the center k which acts on the set of (non-archimedean) valuations of k. This maps T to T . For Theorem 1 we only need the implication 1 2 (1) =⇒ (3) of the next theorem, however we give the full picture, which seems to be of independent interest. Theorem 9. Let S = G(F)/K be the building or symmetric space corresponding to G(F). The following are equivalent: 1. There exists finite index torsion-free subgroups Ω of G′(R ) 1 1 0,T1 and Ω of G′(R ) such that the manifolds or complexes Ω \S and 2 2 0,T2 1 Ω \S are isomorphic. 2