New Perspectives in Arakelov Geometry

Transcription

1 Centre de Reherhes Mthémtiques CRM Proeedings nd Leture Notes Volume 36, 2004 New Perspetives in Arkelov Geometry Cterin Consni nd Mtilde Mrolli Astrt. In this pper we give unified desription of the Arhimeden nd the totlly split degenerte fiers of n rithmeti surfe, using opertor lgers nd Connes theory of spetrl triples in nonommuttive geometry. 1. Introdution The im of this rtile is to report on some of the reent results otined y the two uthors on nonommuttive interprettion of the totlly split degenerte fiers of n rithmeti surfe. The results stted in the first prt of the pper were nnouned in tlk given y the first uthor t the VII meeting of the CNTA in My 2002 t Montrel. The mteril presented in this note is sed on the ppers [11, 12]. Let X e n rithmeti surfe defined over Spe(Z) (or over Spe(O K ), for numer field K), hving the smooth lgeri urve X /Q s its generi fier. It is well known tht, s Riemnn surfe, X /C dmits lwys uniformiztion y mens of Shottky group Γ. In ft, the presene of this uniformiztion plys fundmentl role in the theory of the fier t infinity of X desried in [11, 22]. In nlogy to Mumford s p-di uniformiztion of lgeri urves (f. [26]), the Riemnn surfe X /C n e interpreted s the oundry t infinity of 3-mnifold X Γ defined s the quotient of the rel hyperoli 3-spe H 3 y the tion of the Shottky group Γ. The spe X Γ ontins in its interior n infinite link of ounded geodesis. Mnin gve n expression for the Arkelov Green funtion on X /C in terms of onfigurtions of geodesis in X Γ, thus interpreting this tngle s the dul grph G of the losed fier t infinity of X. In the first prt of this pper we onentrte on Mnin s desription of suh dul grph nd we exhiit the suspension flow S T of dynmil system T on the limit set of the Shottky group Γ s geometri model for the dul grph G Mthemtis Sujet Clssifition. 58B34, 14G40, 46L55, 11G30. This pper ws prtly written during visits of the seond uthor to Florid Stte University nd University of Toronto. The uthors thnk these institutions for the hospitlity. The first uthor is prtilly supported y NSERC grnt The seond uthor is prtilly supported y the Humoldt Foundtion nd the Germn Government (Sofj Kovlevsky Awrd). The uthors thnk Alin Connes for mny extremely helpful disussions. This is the finl form of the pper Amerin Mthemtil Soiety

2 80 CATERINA CONSANI AND MATILDE MARCOLLI In prtiulr, the first ohomology group of S T determines model of the first ohomology of the dul grph of the fier t infinity. Furthermore, the first (o)homology group of S T rries nturl filtrtion. A ruil feture of this onstrution is the ft (proved in [11, Setion 5]) tht this dynmil ohomology ontins suspe isomorphi to the Arhimeden ohomology of [10]: the group of invrints for the tion of the lol monodromy t infinity. This spe hs lso the orret geometri properties, in order to e interpreted in terms of ohomology theory ssoited to degenertion in neighorhood of rithmeti infinity. Our result identifies suh spe with distinguished suspe of the ohomology of topologil spe onstruted in terms of geodesis in X Γ, our geometri model of the dul grph. Under this identifition, the grded struture ssoited to the filtrtion on the (o)homology of S T orresponds to the grded struture given y Tte twists on the Arhimeden ohomology of [10]. The Cuntz Krieger lger O A ssoited to the shift T desries the ring of funtions on nonommuttive spe, whih represents the quotient of the limit set Λ Γ of the Shottky group, y the tion of Γ. In terms of the geometry of the fier t rithmeti infinity, this spe n e thought of s the set of omponents of the speil fier, or equivlently the verties of the dul grph G, wheres the quotient Λ Γ Γ Λ Γ gives the edges of G. The lger O A rries refined informtion on the tion of the Shottky group Γ on its limit set. In prtiulr, we onstrut spetrl triple for this lger. In nonommuttive geometry, the notion of spetrl triple provides the orret generliztion of the lssil struture of Riemnnin mnifold. The two notions gree on ommuttive spe. In the usul ontext of Riemnnin geometry, the definition of the infinitesiml element ds on smooth spin mnifold n e expressed in terms of the inverse of the lssil Dir opertor D. This is the key remrk tht motivtes the theory of spetrl triples. In prtiulr, the geodesi distne etween two points on the mnifold is defined in terms of D 1 (f. [8, Setion VI]). The spetrl triple tht desries lssil Riemnnin spin mnifold is (A, H, D), where A is the lger of omplex vlued smooth funtions on the mnifold, H is the Hilert spe of squre integrle spinor setions, nd D is the lssil Dir opertor ( squre root of the Lplin). These dt determine ompletely nd uniquely the Riemnnin geometry on the mnifold. It turns out tht, when expressed in this form, the notion of spetrl triple extends to more generl nonommuttive spes, where the dt (A, H, D) onsist of C -lger A (or more generlly of smooth sulger of C -lger) with representtion s ounded opertors on Hilert spe H, nd n opertor D on H tht verifies the min properties of Dir opertor. The notion of smoothness is determined y D: the smooth elements of A re defined y the intersetion of domins of powers of the derivtion given y ommuttor with D. The si geometri struture enoded y the theory of spetrl triples is Riemnnin geometry, ut in more refined ses, suh s Kähler geometry, the dditionl struture n e esily enoded s dditionl symmetries. In our onstrutions, the Dir opertor D is otined from the grding opertor ssoited to filtrtion on the ohins of the omplex tht omputes the dynmil ohomology. The indued opertor on the suspe identified with the Arhimeden ohomology grees with the logrithm of Froenius of [10, 15].

3 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 81 This struture further enrihes the geometri interprettion of the Arhimeden ohomology, giving it the mening of spinors on nonommuttive mnifold, with the logrithm of Froenius introdued in [15] in the role of the Dir opertor. An dvntge of this onstrution is tht ompletely nlogous formultion exists in the se of Mumford urves. This provides unified desription of the Arhimeden nd totlly split degenerte fiers of n rithmeti surfe. Let p e finite prime where X hs totlly split degenerte redution. Then, the ompletion X p t p of the generi fier of X is split-degenerte stle urve over Q p (lso lled Mumford urve) uniformized y the tion of p-di Shottky group Γ. The dul grph of the redution of Xp oinides with finite grph otined s the quotient of tree Γ y the tion of Γ. The urve X p is holomorphilly isomorphi to quotient of suset of the ends of the Bruht Tits tree of Q p y the tion of Γ. Thus, in this setting, the Bruht Tits tree t p reples the hyperoli spe H 3 t infinity, nd the nlog of the tngle of ounded geodesis in X Γ is plyed y douly infinite wlks in Γ /Γ. In nlogy with the Arhimeden onstrution, we define the system (W( Γ /Γ), T ) where T is n invertile shift mp on the set W( Γ /Γ) of douly-infinite wlks on the grph Γ /Γ. The first ohomology group H 1 (W( Γ /Γ) T, Z) of the mpping torus W( Γ /Γ) T of T inherits nturl filtrtion using whih we introdue dynmil ohomology group. We gin hve Cuntz-Krieger grph lger C ( Γ /Γ) nd we n onstrut spetrl triple s in the se t infinity, where gin the Dir opertor is relted to the grding opertor Φ tht omputes the lol ftor s regulrized determinnt, s in [16, 17]. In [12], we lso suggested possile wy of extending suh onstrution to ples tht re not of split degenerte redution, inspired y the fom spe onstrution of [5, 21]. 2. Nottion Throughout this pper we will denote y K one mong the following fields: () the omplex numers C, () finite extension of Q p. When () ours, we write O K for the ring of integers of K, m O K for the mximl idel nd π m for uniformizer (i.e., m = (π)). We lso denote y k the residue lsses field k = O/m. We denote y H 3 the three-dimensionl rel hyperoli spe i.e., the quotient H 3 = SU(2)\ PGL(2, C). This spe n lso e desried s the upper hlf spe H 3 C R + endowed with the hyperoli metri. The group PSL(2, C) ts on H 3 y isometries. The omplex projetive line P 1 (C) is identified with the onforml oundry t infinity of H 3 nd the tion of PSL(2, C) on H 3 extends to n tion on H 3 := H 3 P 1 (C). The group PSL(2, C) ts on P 1 (C) y frtionl liner trnsformtions. For n integer g 1, Shottky group of rnk g is disrete sugroup Γ PSL(2, K), whih is purely loxodromi nd isomorphi to free group of rnk g. We denote y Λ Γ the limit set of the tion of Γ. One sees tht Λ Γ is ontined in P 1 (K). This set n lso e desried s the losure of the set of the

4 82 CATERINA CONSANI AND MATILDE MARCOLLI ttrtive nd repelling fixed points z ± (g) of the loxodromi elements g Γ. In the se g = 1 the limit set onsists of two points, ut for g 2 the limit set is usully frtl of some Husdorff dimension 0 δ = dim H (Λ Γ ) < 2. We denote y Ω Γ = Ω Γ (K) the domin of disontinuity of Γ, tht is, the omplement of Λ Γ in P 1 (K). When K = C, the quotient spe X Γ := H 3 /Γ is topologilly hndleody of genus g, nd the quotient X /C = Ω Γ /Γ is Riemnn surfe of genus g. The overing Ω Γ X /C is lled Shottky uniformiztion of X /C. Every omplex Riemnn surfe X /C dmits Shottky uniformiztion. The hndleody X Γ n e omptified y dding the onforml oundry t infinity X /C to otin X Γ := X Γ X /C = (H 3 Ω Γ )/Γ. A direted grph E onsists of dt E = (E 0, E 1, E+, 1 r, s, ι), where E 0 is the set of verties, E 1 is the set of oriented edges w = {e, ɛ}, where e is n edge of the grph nd ɛ = ±1 is hoie of orienttion. The set E+ 1 onsists of hoie of orienttion for eh edge, nmely one element in eh pir {e, ±1}. The mps r, s: E 1 E 0 re the rnge nd soure mps, nd ι is the involution on E 1 defined y ι(w) = {e, ɛ}. A direted grph is finite if E 0 nd E 1 re finite sets. It is lolly finite if eh vertex emits nd reeives t most finitely mny oriented edges in E 1. A vertex v in direted grph is sink if there is no edge in E+ 1 with soure v. A juxtposition of oriented edges w 1 w 2 is sid to e dmissile if w 2 ι(w 1 ) nd r(w 1 ) = s(w 2 ). A (finite, infinite, douly infinite) wlk in direted grph E is n dmissile (finite, infinite, douly infinite) sequene of elements in E 1. We denote y W n (E) the set of wlks of length n, y W (E) = n Wn (E), y W + (E) the set of infinite wlks, nd y W(E) the set of douly infinite wlks. A direted grph is direted tree if, for ny two verties, there exists unique wlk in W (E) onneting them. The edge mtrix A + of lolly finite (or row finite) direted grph is n (#E+) 1 (#E+) 1 (possily infinite) mtrix. The entries stisfy A + (w i, w j ) = 1 if w i w j is n dmissile pth, nd A + (w i, w j ) = 0 otherwise. The direted edge mtrix of E is #E 1 #E 1 (possily infinite) mtrix with entries A(w i, w j ) = 1 if w i w j is n dmissile wlk nd A(w i, w j ) = 0 otherwise. Even when not expliitly stted, ll Hilert spes nd lgers of opertors we onsider will e seprle, i.e., they dmit dense (in the norm topology) ountle suset. 3. A dynmil theory t infinity Let Γ PSL(2, C) e Shottky group of rnk g 2. Given hoie of set of genertors {g i } g i=1 for Γ, there is ijetion etween the elements of Γ nd the set of ll dmissile wlks in the Cyley grph of Γ, nmely redued words in the {g i } 2g i=1, where we use the nottion g i+g := g 1 i, for i = 1,..., g. In the following we onsider the sets S + nd S of resp. right-infinite, douly infinite dmissile sequenes in the {g i } 2g i=1 : (3.1) S + = { l... i {g i } 2g i=1, i+1 1 i, i N}, (3.2) S = {... m l... i {g i } 2g i=1, i+1 1 i, i Z}. The dmissiility ondition simply mens tht we only llow redued words in the genertors, without nelltions.

5 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 83 On the spe S we onsider the topology generted y the sets W s (x, l) = {y S x k = y k, k l}, nd W u (x, l) = {y S x k = y k, k l} for x S nd l Z. There is two-sided shift opertor T ting on S s the mp (3.3) T (... m l... ) =... m l+1... Definition 3.1. A sushift of finite type (S A, T ) onsists of ll douly infinite sequenes in the elements of given finite set W (lphet) with the dmissiility ondition speified y #W #W elementry mtrix, S A = {(w k ) k Z : w k W, A(w k, w k+1 ) = 1}, nd with the tion of the invertile shift (T w) k = w k+1. Lemm 3.2. The spe S n e identified with the sushift of finite type S A with the symmetri 2g 2g mtrix A given y the direted edge mtrix of the Cyley grph of Γ. The two-sided shift opertor T on S of (3.3) deomposes S in produt of expnding nd ontrting diretions, so tht (S, T ) is Smle spe. The following topologil spe is defined in terms of the Smle spe (S, T ) nd will e onsidered s geometri reliztion of the dul grph ssoited to the fier t rithmeti infinity of the rithmeti surfe X. Definition 3.3. The mpping torus (suspension flow) of the dynmil system (S, T ) is defined s (3.4) S T := S [0, 1]/(x, 0) (T x, 1). The spe S T is very nturl spe ssoited to the nonommuttive spe (3.5) Λ Γ Γ Λ Γ S/Z, with Z ting vi the invertile shift T of (3.3), nmely the homotopy quotient (f. [3, 6]), (3.6) S T = S Z R. Nmely, it is ommuttive spe tht provides, up to homotopy, geometri model for (3.5), where the nonommuttive spe (3.5) n e identified with the quotient spe of folition (3.6) whose generi lef is ontrtile ( opy of R) (Co)homology of S T. In this prgrph we give n expliit desription of the (o)homology H 1 (S T, Z). The shift T ting on S indues n utomorphism of the C -lger of ontinuous funtions C(S). With n use of nottion we still denote it y T. Consider the rossed produt C -lger C(S) T Z. This is suitle norm ompletion of C(S)[T, T 1 ] with produt (V W ) k = r Z V k (T r W r+k ), for V = k V kt k, W = k W kt k, nd V W = k (V W ) kt k. The K-theory group K 0 (C(S) T Z) is desried y the o invrints of the tion of T (f. [4, 27]). Theorem 3.4. The ohomology H 1 (S T ) stisfies the following properties (1) There is n identifition of H 1 (S T, Z) with the K 0 -group of the rossed produt C -lger for the tion of T on S, (3.7) H 1 (S T, Z) = K 0 (C(S) T Z).

6 84 CATERINA CONSANI AND MATILDE MARCOLLI (2) The identifition (3.7) endows H 1 (S T, Z) with filtrtion y free elin groups F 0 F 1... F n, with rnk F 0 = 2g nd rnk F n = 2g(2g 1) n 1 (2g 2) + 1, for n 1, so tht H 1 (S T, Z) = lim n F n. (3) This filtrtion is indued y filtrtion P n on the lolly onstnt funtions C(S +, Z) whih depend on future oordintes, with P n given y lolly onstnt funtions tht depend only on the first n + 1 oordintes. The Pimsner Voiulesu ext sequene for the K-theory (3.7) is, (3.8) 0 Z C(S, Z) δ=1 T C(S, Z) H 1 (S T, Z) 0. (4) The ohomology H 1 (S T, κ) = H 1 (S T, Z) κ, for κ = R or C, is omputed y (3.9) 0 C P κ δ P κ H 1 (S T, C) 0. The vetor spe P κ, for P = C(S +, Z) = lim n P n, dmits Hilert spe ompletion L = L 2 (Λ Γ, µ), where µ is the Ptterson Sullivn mesure on the limit set Λ Γ (f. [31]) stisfying (3.10) (γ dµ)(x) = γ (x) δ H dµ(x), γ Γ. Notie tht the djoint δ in the L 2 -inner produt of the ooundry δ of the ohomology H 1 (S T ) is n importnt opertor ssoited to the dynmis of the (one sided) shift T on the limit set Λ Γ, nmely, the opertor 1 R, where R is the Perron Froenius opertor of T. This is the nlog of the Guss Kuzmin opertor studied in [24, 25] in the se of modulr urves. For simpliity of nottion, in the following we will use the sme nottion P nd P n for the Z-modules nd for the κ-vetor spes. It will e ler from the ontext whih one we refer to. We identify the vetor spes P n with finite-dimensionl suspes of L 2 (Λ Γ, µ), y identifying lolly onstnt funtions on S + with lolly onstnt funtions on Λ Γ. The following result omputes the first homology of S T. Proposition 3.5. The homology group H 1 (S T, Z) hs filtrtion y free elin groups K N, (3.11) H 1 (S T, Z) = lim N K N, with K N = rnk(k N ) = The group H 1 (S T, Z) n lso e written s { (2g 1) N + 1 N even (2g 1) N + (2g 1) N odd H 1 (S T, Z) = N=0 R N where R n is free elin group of rnks R 1 = 2g nd R N = rnk(r N ) = 1 µ(d) (2g 1) N/d, N d N

7 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 85 for N > 1, with µ the Möius funtion. This is isomorphi to free elin group on ountly mny genertors. Thus, the Z-modules F n re otined s quotients P n /δp n 1, forδ(f) = f f T, where P = n P n is the module of ontinuous Z-vlued funtions on S tht depend only on future oordintes. The Z-modules K N re generted y ll dmissile words 0... N suh tht the word N 0 is lso dmissile. Comining Theorem 3.4 with Proposition 3.5, we n ompute expliitly the piring of homology nd ohomology for S T. Proposition 3.6. Let F n nd K N e the filtrtions defined, respetively, in Theorem 3.4 nd Proposition 3.5. There is piring (3.12), : F n K N Z [f], x = N f( x), with x = 0... N. Here the representtive f [f] is funtion tht depends on the first n + 1 terms 0... n of sequenes in S, nd x is the truntion of the periodi sequene 0... N fter the first n terms. This piring desends to the diret limits of the filtrtions, where it grees with the lssil ohomology/homology piring (3.13), : H 1 (S T, Z) H 1 (S T, Z) Z Dynmil (o)homology. We define the dynmil ohomology Hdyn 1 s the grded vetor spe given y the sum of the grded piees of the filtrtion of H 1 (S T ), introdued in Theorem 3.4. These grded piees Gr n re onsidered with oeffiients in the nth Hodge Tte twist R(n), for n Z. Similrly, we define the dynmil homology H dyn 1 s the grded vetor spe given y the sum of the terms in the filtrtion of H 1 (S T ), introdued in Proposition 3.5. These vetor spes re gin onsidered with twisted R(n)-oeffiients. The pir (3.14) H 1 dyn H dyn 1 provides geometri setting, defined in terms of the dynmis of the shift opertor T, whih ontins opy of the Arhimeden ohomology of [10] nd of its dul. Definition 3.7. Let H 1 (S T, κ) = lim n F n, for filtrtion F n s in Theorem 3.4, with rel or omplex oeffiients. Let Gr n = F n /F n 1 e the orresponding grded piees, with Gr 0 = F 0. (1) We define grded liner suspe V of the Hilert spe L, s the spn of the elements Π n χ S + (w n,k ), with χ S+ (w n,k ) the hrteristi funtion of S+ (w n,k ) S + with w n,k := n 1 = g k g k... g }{{ k. } n times The opertor Π n is the projetion Π n = Π n Π n 1, with Π n the orthogonl projetion of L onto P n. (2) We define the dynmil ohomology s (3.15) H 1 dyn := n 0 gr Γ 2n H 1 dyn,

8 86 CATERINA CONSANI AND MATILDE MARCOLLI where we set (3.16) gr Γ 2n H 1 dyn := Gr n R R(n) with R(n) = (2π 1) n R. Furthermore, we define the grded suspe of H 1 dyn V := n 0 gr Γ 2n V, where gr Γ 2n V is generted y the elements (2π 1) n χ n+1,k, for χ n,k := [χ S+ (w n,k ) ] Gr n 1. (3) The dynmil homology H dyn 1 is defined s (3.17) H dyn 1 := gr Γ 2n H dyn 1, n 1 where we set (3.18) gr Γ 2nH dyn 1 := K n 1 R R(n). We lso define W H dyn 1 s the grded suspe W = n 1 grγ 2n W, where gr Γ 2n W is generted y the 2g elements (2π 1) n g k g k... g }{{ k. } n times The hoie of indexing the grding y gr2n Γ insted of grn Γ is motivted y omprison to the grding on the ohomologil onstrution of [10]. In [11], we showed tht the suspes V nd V relize opies of the Arhimeden ohomology of [10] emedded in the spe of ohins L of the dynmil ohomology nd in the dynmil ohomology itself, while the pir V W relizes opy of the ohomology of the one of the lol monodromy mp N of [10] inside the pir of dynmil ohomology nd homology Hdyn 1 Hdyn 1. The isomorphism etween V nd the Arhimeden ohomology is relized y nturl hoie of sis of holomorphi differentils for the Arhimeden ohomology, onstruted from the dt of the Shottky uniformiztion s in [22]. An expliit geometri desription for the spe V in terms of geodesis in X Γ is otined y interpreting the hrteristi funtion χ S + (w n,k ) s the est pproximtion within P n to distriution supported on the periodi sequene of period g k, f(g k g k g k g k g k... ) = 1 nd f( ) = 0 otherwise. Suh periodi sequene g k g k g k g k g k... desries the losed geodesi in X Γ tht is the oriented ore of one of the hndles in the hndleody. Thus, the suspe gr Γ 2n V P n is spnned y the est pproximtions within P n to ohomology lsses supported on the ore hndles of the hndleody. In other words, this interprettion views the index n 0 of the grded struture V = n grγ 2n V s mesure of zooming in, with inresing preision for lrger n, on the ore hndles of the hndleody X Γ A spetrl triple from dynmis. Rell tht spetrl triple onsists of the following dt (f. [9]).

9 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 87 Definition 3.8. spetrl triple (A, H, D) onsists of C -lger A with representtion ρ: A B(H) s ounded opertors on Hilert spe H, nd n opertor D (lled the Dir opertor) on H, whih stisfies the following properties: (1) D is self-djoint. (2) For ll λ / R, the resolvent (D λ) 1 is ompt opertor on H. (3) The ommuttor [D, ] is ounded opertor on H, for ll A 0, dense involutive sulger of A. We onsider the Cuntz Krieger lger O A (f. [13] [14]) defined s the universl C -lger generted y prtil isometries S 1,..., S 2g, stisfying the reltions (3.19) S j Sj = I (3.20) j S i S i = j A ij S j S j, where A = (A ij ) is the 2g 2g trnsition mtrix of the sushift of finite type (S, T ), nmely the mtrix whose entries re A ij = 1 whenever i j g, nd A ij = 0 otherwise. The lger O A n e lso desried in terms of the tion of the free group Γ on its limit set Λ Γ (f. [28, 30]), so tht we n regrd O A s nonommuttive spe repling the lssil quotient Λ Γ /Γ. In ft, the tion of Γ on Λ Γ P 1 (C) determines unitry representtion of O A on the Hilert spe L 2 (Λ Γ, µ), given y (3.21) (T γ 1f)(x) := γ (x) δ H/2 f(γx), nd (P γ f)(x) := χ γ (x)f(x), where δ H is the Husdorff dimension of Λ Γ nd the element γ Γ is identified with redued word in the genertors {g j } g j=1 nd their inverses, nd χ γ is the hrteristi funtion of the ylinder Λ Γ (γ) of ll (right) infinite redued words tht egin with the word γ. This determines n identifition of O A with the (redued) rossed produt C -lger, O A = C(ΛΓ ) Γ. We then onsider on the Hilert spe L = L 2 (Λ Γ, µ), the unounded liner self djoint opertor D : L L given y the grding opertor of the filtrtion P n, nmely, (3.22) D = n n Π n. The restrition of this opertor to the suspe V of the dynmil ohomology, isomorphi to the Arhimeden ohomology of [10], grees with the Froenius opertor Φ onsidered in [10], whih omputes the lol ftor s regulrized determinnt s in [15]. We extend the opertor (3.22) to n opertor D on H = L L s (3.23) D L 0 = n (n + 1)( Π n 0) D 0 L = n n(0 Π n ). The presene of shift y one in the grding opertor reflets the shift y one in the grding tht ppers in the dulity isomorphisms on the ohomology of the one of the monodromy mp N s in Proposition 4.8 of [10]. The Dir opertor (3.23) tkes into ount the presene of this shift.

10 88 CATERINA CONSANI AND MATILDE MARCOLLI There is nother possile nturl hoie for the sign of the Dir opertor, insted of the one in (3.23). Insted of eing determined y the sign of the opertor Φ on the Arhimeden ohomology nd its dul, this other hoie is determined y the dulity mp tht exists on the omplex of [10], relized y powers of the monodromy mp (f. [10, Proposition 4.8]). In this se, the sign would then e given y the opertor ( ) 0 1 F =, 1 0 tht exhnges the two opies of L in H. We ssume tht the Shottky group Γ hs limit set Λ Γ of Husdorff dimension δ H < 1. Then the dt ove define the dynmil spetrl triple t rithmeti infinity. Theorem 3.9. The dt (O A, H, D), where the lger O A ts digonlly on H = L L, nd the opertor D is given y (3.23) form spetrl triple in the sense of Connes, s in Definition 3.8. The ound on the ommuttors with the genertors S i of O A nd their djoints is otined in [11] in terms of the Poinré series of the Shottky group. The spetrl triple defined this wy ppers to e relted to spetrl triples for AF lgers, reently introdued in [1]. In suh onstrutions the Dir opertor generlizes the grding opertor (3.22), y opertors of the form D = α n Πn, where Π n = Π n Π n 1 re the projetions ssoited to filtrtion of the AF lger, nd the oeffiients α n given y sequene of positive rel numers, stisfying ertin growth onditions. The C -lger C(Λ Γ ) is ommuttive AF-lger (pproximtely finitedimensionl), otined s the diret limit of the finite-dimensionl ommuttive C -lgers generted y hrteristi funtions of overing of Λ Γ. This gives rise to the filtrtion P n in Theorem 3.4, hene our hoie of Dir opertor fits into the setting of [1] for the AF lger C(Λ Γ ). On the other hnd, while in the onstrution of [1] the eigenvlues α n n e hosen suffiiently lrge, so tht the resulting spetrl triple for the AF lger would e finitely summle, when we onsider the Cuntz Krieger lger O A = C(ΛΓ ) Γ, the oundedness of ommuttors (ondition 3 of Definition 3.8) n only e stisfied for the speil hoie of α n = n, with onstnt, f. [1, Remrk 2.2], whih does not yield finitely summle spetrl triple. The reson for this lies in well known result of Connes [7] whih shows tht non menle disrete groups (s is the se for the Shottky group Γ) do not dmit finitely summle spetrl triples. Thus, if the dense sulger of O A with whih D hs ounded ommuttors ontins group elements, then the Dir opertor D nnot e finitely summle. In our onstrution, the hoie of sign for D is presried y the grded struture of the ohomology theory of [10] nd y the identifition of the Arhimeden ohomology of [10] nd its dul with suspes of the dynmil ohomology nd homology s in [11]. This wy, the requirement tht the Dir opertor grees with the opertor Φ of [10] on these suspes fixes the hoie of the sign of the Dir opertor, whih rries the topologil informtion on the nonommuttive mnifold. In the onstrution of spetrl triples for AF lgers of [1] only the metri spet of the spetrl triple is retined, tht is, the opertor onsidered is of the form D, while the sign is not disussed.

11 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 89 A possile wy to refine the onstrution of the spetrl triple nd del with the lk of finite summility is through the ft tht the Cuntz Krieger lger O A hs seond desription s rossed produt lger. Nmely, up to stiliztion (i.e., tensoring with ompt opertors) we hve (3.24) O A F A T Z, where F A is n pproximtely finite-dimensionl (AF) lger, f. [13, 14]. This lger n e desried in terms of groupoid C -lger ssoited to the unstle mnifold in the Smle spe (S, T ). In ft, onsider the lger O lg A generted lgerilly y the S i nd Si sujet to the Cuntz Krieger reltions (3.19) (3.20). Elements in O lg A re liner omintions of monomils S µsν, for multi-indies µ, ν, f. [14]. The AF lger F A is generted y elements S µ Sν with µ = ν, nd is filtered y finite-dimensionl lgers F A,n generted y elements of the form S µ P i Sν with µ = ν = n nd P i = S i Si the rnge projetions, nd emeddings determined y the mtrix A. The ommuttive lger C(Λ Γ ) sits s sulger of F A generted y ll rnge projetions S µ Sµ. The emedding is omptile with the filtrtion nd with the tion of the shift T, whih is implemented on F A y the trnsformtion i S isi. (f. [14].) The ft tht the lger n e written in the form (3.24) implies tht, y Connes result on hyperfiniteness [7], it my rry finitely summle spetrl triple. It is n interesting prolem whether the onstrution of finitely summle triple n e rried out in wy tht is of rithmeti signifine Lol ftor. For n rithmeti vriety X over Spe Z, the Arhimeden ftor (lol ftor t rithmeti infinity) L κ (H m, s) is produt of Gmm funtions, with exponents nd rguments tht depend on the (pure) Hodge struture H m = H m (X, C) = p+q=m Hp,q. More preisely, (f. [29]) { (3.25) L κ (H m p,q, s) = Γ C(s min(p, q)) hp,q κ = C p<q Γ C(s p) hp,q p Γ R(s p) hp+ Γ R (s p+1) hp κ = R, where the h p,q, with p + q = m, re the Hodge numers, h p,± is the dimension of the ±( 1) p -eigenspe of de Rhm onjugtion on H p,p, nd Γ C (s) := (2π) s Γ(s) Γ R (s) := 2 1/2 π s/2 Γ(s/2). Deninger produed unified desription of the ftors t rithmeti infinity nd t the finite primes, in the form of Ry Singer determinnt (f. [15]). The ftor (3.25) stisfies (3.26) L κ (H m, s) = det ( ) 1 1 2π (s Φ) V m, where V m is n infinite-dimensionl rel vetor spe. The zet regulrized determinnt of n unounded self djoint opertor T is defined s det (s T ) = exp( dζ T (s, z)/dz z=0 ). In [10], the grded spes V re identified with inerti invrints of doule omplex of rel Tte-twisted differentil forms on X /C with suitle utoffs. Nmely, suh omplex is endowed with the tion of n endomorphism N, whih represents logrithm of the lol monodromy t rithmeti infinity, nd the spes V re identified with the kernel of the mp N on the hyperohomology. In prtiulr, in

12 90 CATERINA CONSANI AND MATILDE MARCOLLI the se of n rithmeti surfe, we showed in [11] tht the Arhimeden ohomology group V 1 is identified with the suspe V of P nd with the orresponding suspe of the dynmil ohomology H 1 dyn. The dynmil spetrl triple of Theorem 3.9 is not finitely summle. However, it is possile to reover from these dt the lol ftor t rithmeti infinity (3.26) for m = 1. Proposition Consider the zet funtions (3.27) ζ π(v),d (s, z) := Tr ( π(v)π(λ, D) ) (s λ) z, λ Spe(D) for π(v) the orthogonl projetion on the norm losure of V in L, nd (3.28) ζ π(v,f =id),d (s, z) := λ Spe(D) Tr ( π(v, F = id)π(λ, D) ) (s λ) z, for π(v, F = id) the orthogonl projetion on the norm losure of V F =id. The orresponding regulrized determinnts stisfy ( exp d ) 1 (3.29) dz ζ π(v),d/2π(s/2π, z) z=0 = L C (H 1 (X), s), (3.30) ( exp d 1 dz ζ π(v,f (s/2π, z) =id),d/2π z=0) = L R (H 1 (X), s). Moreover, the opertor π(v) ts on the rnge of the spetrl projetions Π(λ, D) s ertin elements of the lger O A. Here F is the involution indued y the rel struture on X /R, whih orresponds to the hnge of orienttion on the geodesis in X Γ nd on S T. 4. A dynmil theory for Mumford urves Throughout this hpter K will denote finite extension of Q p nd K the Bruht-Tits tree ssoited to G = PGL(2, K). In the following we rell few results out the tion of Shottky group on Bruht-Tits tree nd on C -lgers of grphs. Detiled explntions re ontined in [2, 18 20, 26, 30]. Rell tht the Bruht Tits tree is onstruted s follows. One onsiders the set of free O-modules of rnk 2: M V. Two suh modules re equivlent M 1 M 2 if there exists n element λ K, suh tht M 1 = λm 2. The group GL(V ) of liner utomorphisms of V opertes on the set of suh modules on the left: gm = {gm m M}, g GL(V ). Notie tht the reltion M 1 M 2 is equivlent to the ondition tht M 1 nd M 2 elong to the sme orit of the enter K GL(V ). Hene, the group G = GL(V )/K opertes (on the left) on the set of lsses of equivlent modules. We denote y 0 K the set of suh lsses nd y {M} the lss of the module M. Beuse O is prinipl idels domin nd every module M hs two genertors, it follows tht {M 1 }, {M 2 } 0 K, M 1 M 2 = M 1 /M 2 O/m l O/m k, l, k N. The multiplition of M 1 nd M 2 y elements of K preserves the inlusion M 1 M 2, hene the nturl numer (4.1) d({m 1 }, {M 2 }) = l k

13 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 91 Figure 1. The grphs Γ /Γ for genus g = 2, nd the orresponding fiers. is well defined. The grph K of the group PGL(2, K) is the infinite grph with set of verties 0 K, in whih two verties {M 1}, {M 2 } re djent nd hene onneted y n edge if nd only if d({m 1 }, {M 2 }) = 1. (f. [20, 26].) For Shottky group Γ PGL(2, K) there is smllest sutree Γ K ontining the xes of ll elements of Γ. The set of ends of Γ in P1 (K) is Λ Γ, the limit set of Γ. The group Γ rries Γ into itself so tht the quotient Γ /Γ is finite grph tht oinides with the dul grph of the losed fire of the miniml smooth model of the lgeri urve C/K holomorphilly isomorphi to X Γ := Ω Γ /Γ (f. [26, p. 163]). There is smllest tree Γ on whih Γ ts nd suh tht Γ /Γ is the (finite) grph of the speiliztion of C. The urve C is k-split degenerte, stle urve. When the genus of the fiers is t lest 2 i.e., when the Shottky group hs t lest g 2 genertors the urve X Γ is lled Shottky Mumford urve. The possile grphs Γ /Γ nd the orresponding fier for the se of genus 2 re illustrted in Figure 1. To lolly finite direted grph one ssoites C -lger in the following wy. A Cuntz Krieger fmily onsists of olletion {P v } v E 0 of mutully orthogonl projetions nd {S w } w E 1 + of prtil isometries, stisfying the onditions: SwS w = P r(w) nd, for ll v s(e+), 1 P v = w:s(w)=v S wsw. The Cuntz Krieger elements {P v, S w } stisfy the reltion S ws w = A + (w, w)s w S w, with A + the edge mtrix of the grph. One defines universl C -lger C (E) generted y Cuntz Krieger fmily. If E is finite grph with no sinks, we hve C (E) O A+, where O A+ is the Cuntz Krieger lger of the edge mtrix A +. If the direted grph is tree, then C ( ) is n AF lger strongly Morit equivlent to the ommuttive C -lger C 0 ( ). A monomorphism of direted trees indues n injetive -homomorphism of the orresponding C -lgers.

14 92 CATERINA CONSANI AND MATILDE MARCOLLI If G Aut(E) is group ting freely on the direted grph E, with quotient grph E/G, then the rossed produt C -lger C (E) G is strongly Morit equivlent to C (E/G). In prtiulr, if is the universl overing tree of direted grph E nd Γ is the fundmentl group, then the lger C (E) is strongly Morit equivlent to C 0 ( ) Γ. In the following we onsider the Bruht Tits tree K for fixed finite extension K of Q p, nd the orresponding C -lger C ( K ), whih is strongly Morit equivlent to the elin C -lger of omplex vlued funtions C(P 1 (K)). Notie tht the ommuttive C -lger C (X K ) of omplex vlued ontinuous funtions on the Mumford urve X K = Ω Γ /Γ is strongly Morit equivlent to the rossed produt C 0 (Ω Γ ) Γ, with C 0 (Ω Γ ) Γ-invrint idel of C ( P 1 (K) ) with quotient lger C(Λ Γ ). The lger C ( Γ /Γ), in turn, is strongly Morit equivlent to C ( Γ ) Γ nd to C(Λ Γ ) Γ, where Γ = Λ Γ. Similrly, one sees tht the lger C ( K /Γ) is strongly Morit equivlent to the rossed produt lger C ( K ) Γ, whih in turn is strongly Morit equivlent to C ( P 1 (K) ) Γ. Thus, up to Morit equivlene, the grph lger C ( K /Γ) n e regrded s wy of extending the ommuttive C -lger C (X K ) (funtions on the Mumford urve) y the Cuntz-Krieger lger C ( Γ /Γ) ssoited to the edge mtrix of the finite grph Γ /Γ. In this prgrph we introdue dynmil system ssoited to the spe W( /Γ) of wlks on the direted tree on whih Γ ts. In prtiulr, we re interested in the ses when = K, Γ. For = Γ, we otin sushift of finite type ssoited to the tion of the Shottky group Γ on the limit set Λ Γ, of the type tht ws onsidered in [11]. Let V Γ e finite sutree whose set of edges onsists of one representtive for eh Γ-lss. This is fundmentl domin for Γ in the wek sense (following the nottion of [20]), sine some verties my e identified under the tion of Γ. Correspondingly, V P 1 (K) is the set of ends of ll infinite pths strting t points in V. Consider the set W( Γ /Γ) of douly infinite wlks on the finite grph Γ /Γ. These re douly infinite dmissile sequenes in the finite lphet given y the edges of V with oth possile orienttions. On W( Γ /Γ) we onsider the topology generted y the sets W s (ω, l) = { ω W( Γ /Γ) : ω k = ω k, k l} nd W u (ω, l) = { ω W( Γ /Γ) : ω k = ω k, k l}, for ω W( Γ /Γ) nd l Z. With this topology, the spe W( Γ /Γ) is totlly disonneted ompt Husdorff spe. The invertile shift mp T, given y (T ω) k = ω k+1, is homeomorphism of W( Γ /Γ). We n desrie gin the dynmil system (W( Γ /Γ), T ) in terms of sushifts of finite type. Lemm 4.1. The spe W( Γ /Γ) with the tion of the invertile shift T is sushift of finite type, where W( Γ /Γ) = S A with A the direted edge mtrix of the finite grph Γ /Γ. We onsider the mpping torus of T : (4.2) W( Γ /Γ) T := W( Γ /Γ) [0, 1]/(T x, 0) (x, 1) Genus two exmple. In the exmple of Mumford Shottky urves of genus g = 2, the tree Γ is illustrted in Figure 2. In the first se in the figure, the tree Γ is just opy of the Cyley grph of the free group Γ on two genertors, hene we n identify douly infinite wlks

15 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 93 Figure 2. The grphs Γ /Γ for genus g = 2, nd the orresponding trees Γ. in Γ with douly infinite redued words in the genertors of Γ nd their inverses. The direted edge mtrix is given y A = In the seond se in Figure 2, we lel y = e 1, = e 2 nd = e 3 the oriented edges in the grph Γ /Γ, so tht we hve orresponding set of lels E = {,,, ā,, } for the edges in the overing Γ. A hoie of genertors for the group Γ Z Z ting on Γ is otined y identifying the genertors g 1 nd g 2 of Γ with the hins of edges nd. Douly infinite wlks in the tree Γ re dmissile douly infinite sequenes of suh lels, where dmissiility is

16 94 CATERINA CONSANI AND MATILDE MARCOLLI determined y the direted edge mtrix A = The third se in Figure 2 is nlogous. A hoie of genertors for the group Γ Z Z ting on Γ is given y ā nd. Douly infinite wlks in the tree Γ re dmissile douly infinite sequenes in the lphet E = {,,, ā,, }, with dmissiility determined y the direted edge mtrix A = The onstrution is nlogous for genus g > 2, for the vrious possile finite grphs Γ /Γ. The direted edge mtrix n then e written in lok form s ( ) α11 α A = 12, α 21 α 22 where eh lok α ij is #( Γ /Γ) 1 + #( Γ /Γ) 1 +-mtrix with α 12 = α12, t α 21 = α21, t nd α 11 = α22. t 4.2. Cohomology of W( /Γ) T. Let = Γ. We identify the first ohomology group H 1 (W( Γ /Γ) T, Z) with the group of homotopy lsses of ontinuous mps of W( Γ /Γ) T to the irle. Let C(W( Γ /Γ), Z) e the Z-module of integer vlued ontinuous funtions on W( Γ /Γ), nd let C(W( Γ /Γ), Z) T := Coker(δ), for δ(f) = f f T. The nlog of Theorem 3.4 holds: Proposition 4.2. The mp f [ exp ( 2πitf(x) )], whih ssoites to n element f C(W( Γ /Γ), Z) homotopy lss of mps from W( Γ /Γ) T to the irle, gives n isomorphism C(W( Γ /Γ), Z) T H 1 (W( Γ /Γ) T, Z). Moreover, there is filtrtion of C(W( Γ /Γ), Z) T y free Z-modules F 0 F 1 F n, of rnk θ n θ n 1 + 1, where θ n is the numer of dmissile words of length n + 1 in the lphet, so tht we hve H 1 (W( Γ /Γ) T, Z) = lim n F n. The quotients F n+1 /F n re lso torsion-free. The spe W( Γ /Γ) T orresponds to spe of ounded geodesis on the grph K /Γ, where geodesis, in this setting, re just douly infinite wlks in K /Γ. In prtiulr, losed geodesi is the imge under the quotient mp π Γ : K K /Γ of douly infinite wlk in the Bruht-Tits tree K with ends given y the pir z + (γ), z (γ) of fixed points of some element γ Γ. Similrly, ounded geodesi is n element ω W( K /Γ) whih is the imge, under the

17 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 95 quotient mp, of douly infinite wlk in K with oth ends on Λ Γ P 1 (K). This implies tht ounded geodesi is wlk of the form ω = π Γ ( ω), for some ω W( Γ /Γ). By onstrution, ny suh wlk is n xis of Γ. Orits of W( Γ /Γ) under the tion of the invertile shift T orrespond ijetively to orits of the omplement of the digonl in Λ Γ Λ Γ under the tion of Γ. Thus, we see tht W( Γ /Γ) T gives geometri reliztion of the spe of ounded geodesis on the grph K /Γ, muh s, in the se of the geometry t rithmeti infinity, we used the mpping torus of the shift T s model of the tngle of ounded geodesis in hyperoli hndleody. As in the se t infinity, we n onsider the Pimsner Voiulesu ext sequene omputing the K-theory groups of the rossed produt C -lger C ( W( Γ /Γ) ) T Z, (4.3) 0 H 0 (W( Γ /Γ) T, Z) C(W( Γ /Γ), Z) In the orresponding sequene δ=1 T C(W( Γ /Γ), Z) H 1 (W( Γ /Γ) T, Z) 0. (4.4) 0 H 0 (W( Γ /Γ) T, κ) P δ P H 1 (W( Γ /Γ) T, κ) 0, for the ohomology for H (W( Γ /Γ) T, κ), with κ = R or C, we n tke the vetor spe P otined, s in the se t infinity, y tensoring with κ the Z-module P C(W( Γ /Γ), Z) of funtions of future oordintes where P C(W + ( Γ /Γ), Z). This hs filtrtion P = n P n, where P n is identified with the sumodule of C(W + ( Γ /Γ), Z) generted y hrteristi funtions of W + ( Γ /Γ, ρ) W + ( Γ /Γ), where ρ W ( Γ /Γ) is finite wlk ρ = w 0... w n of length n+1, nd W + ( Γ /Γ, ρ) is the set of infinite pths ω W + ( Γ /Γ), with ω k = w k for 0 k n + 1. This filtrtion defines the terms F n = P n /δp n 1 in the filtrtion of the dynmil ohomology of the Mumford urve, s in Proposition 4.2. Agin, we will use the sme nottion in the following for the free Z-module P n of funtions of t most n + 1 future oordintes nd the vetor spe otined y tensoring P n y κ. We otin Hilert spe ompletion of the spe P of ohins in (4.4) y onsidering L = L 2 (Λ Γ, µ) defined with respet to the mesure on Λ Γ = Γ given y ssigning its vlue on the lopen set V (v), given y the ends of ll pths in Γ strting t vertex v, to e µ(v (v)) = q d(v) 1, with q = rd(o/m). In [11, Setion 4], we showed how the mpping torus S T of the sushift of finite type (S, T ), ssoited to the limit set of the Shottky group, mps surjetively to the tngle of ounded geodesis inside the hyperoli hndleody, through mp tht resolves ll the points of intersetion of different geodesis. In the se of the Mumford urve, where we reple the rel hyperoli 3-spe y the Bruht Tits uilding K, the nlog of the surjetive mp from S T to the tngle of ounded geodesis is mp from W( Γ /Γ) T to the dul grph Γ /Γ. Here is desription of this mp.

18 96 CATERINA CONSANI AND MATILDE MARCOLLI As efore, we write elements of W( Γ /Γ) s dmissile douly infinite sequenes ω =... w i m... w i 1 w i0 w i1... w in..., with the w ik = {e ik, ɛ ik } oriented edges on the grph Γ /Γ. We onsider eh oriented edge w of normlized length one, so tht it n e prmeterized s w(t) = {e(t), ɛ}, for 0 t 1, with w(t) = {e(1 t), ɛ}. Sine ω S A is n dmissile sequene of oriented edges we hve w ik (1) = w ik+1 (0) (0) Γ. We onsider mp of the overing spe W( Γ /Γ) R of W( Γ /Γ) T to Γ of the form (4.5) Ẽ(ω, τ) = w i[τ] (τ [τ]). Here Γ denotes the geometri reliztion of the grph. By onstrution, the mp Ẽ stisfies Ẽ(T ω, τ) = Ẽ(ω, τ + 1), hene it desends to mp E of the quotient (4.6) E : W( Γ ) T Γ. We then otin mp to Γ /Γ, y omposing with the quotient mp of the Γ tion, π Γ : Γ Γ /Γ, tht is, (4.7) E := π Γ E : W( Γ ) T Γ /Γ. Thus, we otin the following. Proposition 4.3. The mp E of (4.7) is ontinuous surjetion from the mpping torus W( Γ ) T to the geometri reliztion Γ /Γ of the finite grph Γ /Γ. The fiers of the mp (4.7) re expliitly desried s (4.8) E 1 (w i (t)) = W( Γ )(w i ) {t} W( Γ )(w i ) {1 t}, where w i (t), for t [0, 1] is prmeterized oriented edge in the grph Γ /Γ, nd W( Γ )(w i ) W( Γ ) onsists of W( Γ )(w i ) = {ω W( Γ ) : w i0 = w i }, for ω =... w i m... w i 1 w i0 w i1... w in.... We n lso onsider the onstrution desried ove, where, insted of using the tree Γ, we use lrger Γ-invrint trees inside the Bruht Tits tree. The set of douly infinite wlks W( K /Γ) n e identified with the set of dmissile douly infinite sequenes in the oriented edges of fundmentl domin for the tion of Γ. Therefore, we otin the following identifition (4.9) W( K /Γ) V W ( Γ /Γ) V V W + ( Γ /Γ) W + ( Γ /Γ) V W( Γ /Γ), where we distinguish etween wlks tht wnder off from the finite grph Γ /Γ long one of the pths leding to n end in V, in one or in oth diretions, nd those tht sty onfined within the finite grph Γ /Γ. Here we inlude in W ( Γ /Γ) = n W n ( Γ /Γ) lso the se W 0 ( Γ /Γ) =, where the wlk in K /Γ does not interset the finite grph Γ /Γ t ll. In the topology indued y the p-di norm, P 1 (K) is totlly disonneted ompt Husdorff spe, nd so is W( K /Γ) y (4.9). Agin, we onsider the invertile shift mp T, whih is homeomorphism of W( K /Γ), nd we form the mpping torus (4.10) W( K /Γ) T := W( K /Γ) [0, 1]/(T x, 0) (x, 1).

19 NEW PERSPECTIVES IN ARAKELOV GEOMETRY 97 We otin the nlog of Proposition 4.2, though in the se of K the Z-modules F n will not e finitely generted. On the other hnd, we n restrit to neighorhoods of the tree Γ inside K, whih orrespond to the redution mps modulo powers of the mximl idel. In ft, in the theory of Mumford urves, it is importnt to onsider lso the redution modulo powers m n of the mximl idel m O K, whih provides infinitesiml neighorhoods of order n of the losed fier. For eh n 0, we onsider sugrph K,n of the Bruht Tits tree K defined y setting 0 K,n := {v 0 K : d(v, Γ) n}, with respet to the distne (4.1), with d(v, Γ ) := inf{d(v, ṽ) : ṽ ( Γ )0 }, nd 1 K,n := {w 1 K : s(w), r(w) 0 K,n}. Thus, we hve K,0 = Γ. We hve K = n K,n. For ll n N, the grph K,n is invrint under the tion of the Shottky group Γ on, nd the finite grph K,n /Γ gives the dul grph of the redution X K O/m n+1. They form direted fmily with inlusions j n,m : K,n K,m, for ll m n, with ll the inlusions omptile with the tion of Γ. In [12] we introdue the dynmil ohomology nd disuss spetrl geometry for the C -lgers C ( K,n /Γ) C ( K,n ) Γ Spetrl triples nd Mumford urves. We onsider the Hilert spe H = L L nd the opertor D defined s D L 0 = 2π (n + 1) Πn D 0 L = 2π (4.11) n Πn, R log q R log q where Π n = Π n Π n 1 re the orthogonl projetions ssoited to the filtrtion P n, the integer R is the length of ll the words representing the genertors of Γ (this n e tken to e the sme for ll genertors, possily fter lowing up finite numer of points on the speil fier, s explined in [12]), nd q = rd(o/m). The sme rgument used in [11] for the se t rithmeti infinity dpts to the se of Mumford urves to prove the following result. (Note: the sttement elow orrets n unfortunte mistke tht ourred in [12, Setion 5.4].) Theorem 4.4. Consider the tree Γ of the p-di Shottky group ting on K. (1) There is representtion of the lger C ( Γ /Γ) y ounded liner opertors on the Hilert spe L. (2) The dt (C ( Γ /Γ), H, D), with the lger ting digonlly on H = L L, nd the Dir opertor D of (4.11) form spetrl triple. In [12] we reovered rithmeti informtion suh s the lol L-ftors of [16] from the dynmil ohomology nd the dt of the spetrl triple of Theorem 4.4. Rell tht, for urve X over glol field K, ssuming semi-stility t ll ples of d redution, the lol Euler ftor t finite ple v hs the following desription [29]: (4.12) L v (H 1 (X), s) = det(1 Fr v N(v) s H 1 (X, Q l ) Iv ) 1. Here Fr v is the geometri Froenius ting on l-di ohomology of X = X Spe(K), with K n lgeri losure nd l prime with (l, q) = 1, where N(v) is the rdinlity of the residue field k(v) t v. The determinnt is evluted on

20 98 CATERINA CONSANI AND MATILDE MARCOLLI the inerti invrints H 1 (X, Q l ) Iv t v (ll of H 1 (X, Q l ) when v is ple of good redution). Suppose v is ple of k(v)-split degenerte redution. Then the ompletion of X t v is Mumford urve X Γ. In this se, the Euler ftor (4.12) tkes the following form: (4.13) L v (H 1 (X), s) = (1 q s ) g where q = N(v). This ftor is omputed y the zet regulrized determinnt (4.14) det,π(v),id (s) = L v (H 1 (X), s) 1, where (4.15) det,,id (s) := exp ( ζ,id,+(s, 0) ) exp ( ζ,id, (s, 0) ), for (4.16) ζ,id,+ (s, z) := ζ,id, (s, z) := λ Spe(iD) i[0, ) λ Spe(iD) i(,0) Tr(Π λ )(s + λ) z Tr(Π λ )(s + λ) z. The element = π(v) is the projetion onto liner suspe V of H, whih is otined vi emeddings of the ohomology of the dul grph Γ /Γ into the spe of ohins of the dynmil ohomology. The projetion π(v) ts on the rnge of the spetrl projetions Π n of D s elements Q n in the AF lger ore of the C -lger C ( Γ /Γ). Notie how, unlike the lol ftor t infinity, the ftor t the non-arhimeden ples involves the full spetrum of D nd not just its positive or negtive prt. It is elieved tht this differene should orrespond to the presene of n underlying geometri spe sed on loop geometry, whih mnifests itself s loops t the non- Arhimeden ples nd s hlf loops (holomorphi disks) t rithmeti infinity. There is nother differene etween Arhimeden nd non-arhimeden ses. At the Arhimeden prime the lol ftor is desried in terms of zet funtions for Dir opertor D (f. [11, 15]). On the other hnd, t the non-arhimeden ples, in order to get the orret normliztion s in [17], we need to introdue rottion of the Dir opertor y the imginry unit, D id. This rottion orresponds to the Wik rottion tht moves poles on the rel line to poles on the imginry line (zeroes for the lol ftor) nd ppers to e mnifesttion of rottion from Minkowskin to Euliden signture it t, s lredy remrked y Mnin [23, p. 135], who wrote tht imginry time motion my e held responsile for the ft tht zeroes of Γ(s) 1 re purely rel wheres the zeroes of ll non-arhimeden Euler ftors re purely imginry. It is expeted, therefore, tht more refined onstrution would involve version of spetrl triples for Minkowskin signture. Referenes 1. C. Antonesu nd E. Christensen, Spetrl triples for AF C -lgers nd metris on the Cntor set, mth.oa/ T. Btes, D. Psk, I. Reurn, nd W. Szymński, The C -lgers of row finite grphs, New York J. Mth. 6 (2000), P. Bum nd A. Connes, Geometri K-theory for Lie groups nd folitions, Brown University/IHES, 1982 (preprint); Enseign. Mth. (2) 46 (2000), no. 1-2, 3 42.

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