Arithmetic Noncommutative

Geometry

http://dx.doi.org/10.1090/ulect/036

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University

LECTURE Series

Volume 3 6

Arithmetic Noncommutative

Geometry Matilde Marcoll i

with a foreword b y Yuri Manln

American Mathematica l Societ y Providence, Rhode Island

EDITORIAL COMMITTE E

Je r ry L . B o n a (Chair ) Eri c M , Priedlande r Adr iano Gars i a Nige l J . Higso n

Pe te r Landwebe r

2000 Mathematics Subject Classification. P r i m a r y 58B34 .

F i g u r e s 1 , 3 , a n d 9 ( p p . 82 , 84 5 a n d 102 ) ar e u s e d w i t h p e r m i s s i o n .

For addi t iona l informatio n an d u p d a t e s o n th i s book , visi t w w w . a m s . o r g / f o o o k p a g e s / u l e e t ~ 3 6

Library o f Congres s Cataloging- in-Publieat io n D a t a

MarcolM, Matilde . Arithmetic noncommutativ e geometr y / Matild e Mareolli .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v, 36 ) Includes bibliographica l references . ISBN 0-8218-3833- 4 (acid-fre e paper ) 1. Noncommutativ e differentia l geometry . I . Title . II . Universit y lectur e serie s (Providence ,

R.I.) ; 36.

QA641.M34 200 5 516.3/6~~dc22 200504783 3

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Visit th e AM S hom e pag e a t ht tp: / /www.ams.org /

10 9 8 7 6 5 4 3 2 1 1 0 09 0 8 0 7 0 6 0 5

And indee d ther e wil l be tim e To wonder "D o I dare?" and , "D o I dare? " Time t o tur n bac k an d descen d th e stair .

Do I dar e Disturb th e Universe ?

For I have know n the m al l already , know n the m all ; Have know n th e evenings , mornings , afternoons , I hav e measure d ou t m y lif e wit h coffe e spoons .

I shoul d hav e been a pai r o f ragged claw s Scuttling acros s the floor s o f silen t seas .

No! I a m no t Princ e Hamlet , no r wa s mean t t o be ; Am a n attendan t lord , on e tha t wil l d o To swel l a progress , star t a scene o r tw o

At times , indeed , almos t ridiculous -Almost, a t times , th e Fool .

We have lingere d i n the chamber s o f the se a By sea-girl s wreathe d wit h seawee d re d an d brow n Till huma n voice s wake us , an d w e drown .

(T.S. Eliot , "The Love Song of J. Alfred Prufrock")

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Contents

Foreword i x

Chapter 1 . Ouvertur e 1 1. Th e NC G dictionar y 3 2. Noncommutativ e space s 4 3. Spectra l triple s 6 4. Wh y noncommutativ e geometry ? 1 2

Chapter 2 . Noncommutativ e modula r curve s 1 5 1. Modula r curve s 1 5 2. Th e noncommutativ e boundar y o f modula r curve s 2 2 3. Limitin g modula r symbol s 2 7 4. Heck e eigenforms 3 8 5. Selber g zet a functio n 4 1 6. Th e modula r comple x an d if-theor y o f C*-algebra s 4 2 7. Intermezzo : chaoti c cosmolog y 4 4

Chapter 3 . Quantu m statistica l mechanic s an d Galoi s theor y 5 1 1. Quantu m statistica l mechanic s 5 3 2. Th e Bost-Conne s syste m 5 6 3. Noncommutativ e geometr y an d Hilbert' s 12t h proble m 6 1 4. Th e GL 2 syste m 6 4 5. Quadrati c fields 7 0

Chapter 4 . Noncommutativ e geometr y a t arithmeti c infinit y 8 1 1. Schottk y uniformizatio n 8 1 2. Dynamic s an d noncommutativ e geometr y 8 8 3. Arithmeti c infinity : archimedea n prime s 9 3 4. Arakelo v geometr y an d hyperboli c geometr y 9 7 5. Intermezzo : quantu m gravit y an d blac k hole s 10 0 6. Dua l grap h an d noncommutativ e geometr y 10 5 7. Arithmeti c varietie s an d L- factors 10 9 8. Archimedea n cohomolog y 11 5

Chapter 5 . Vista s 12 5

Bibliography 13 1

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Foreword

Noncommutative geometr y nowaday s look s a s a vas t buildin g site .

On the one hand, practitioners of noncommutative geometr y (o r ge-ometries) alread y buil t u p a large and swiftl y growin g body o f exciting mathematics, challengin g traditiona l boundarie s an d subdivisions .

On the othe r hand , noncommutativ e geometr y lack s common foun -dations: fo r many interesting constructions of "noncommutativ e spaces" we cannot eve n sa y fo r sur e whic h o f them lea d t o isomorphi c spaces , because the y ar e no t object s o f an all-embracin g categor y (lik e that o f locally ringed topologica l space s in commutativ e geometry) .

Matilde Marcolli' s lecture s reflec t thi s spiri t o f creative growt h an d interdisciplinary research .

She start s Chapte r 1 with a sketc h o f philosoph y o f noncommuta -tive geometr y a la Alai n Connes . Briefly , Conne s suggest s imaginin g C*-algebras a s coordinat e rings . H e the n supplie s severa l bridge s t o commutative geometr y b y hi s construction o f "ba d quotients " o f com-mutative space s via crossed product s an d hi s treatment o f noncommu -tative Riemannian geometry . Finally , algebrai c tools like X-theory an d cyclic cohomology serv e to furthe r enhanc e geometri c intuition .

Marcolli the n proceed s t o explainin g som e recen t development s drawing upo n he r recen t wor k wit h severa l collaborators . A commo n thread i n all o f them i s the stud y o f various aspect s o f uniformization : classical modula r group , Schottk y groups . Th e modula r grou p act s upon th e comple x hal f plane , partiall y compactifie d b y cusps : rationa l points o f the boundar y projectiv e line . Th e actio n become s "bad " a t irrational points , an d her e i s wher e noncommutativ e geometr y enter s the game . A wealt h o f classica l numbe r theor y i s encode d i n th e co -efficients o f modula r forms , thei r Melli n transforms , Heck e operator s and modula r symbols . Thei r counterpart s livin g a t th e noncommuta -tive boundar y hav e onl y recentl y starte d t o unrave l themselves , an d Marcolli give s a beautifu l overvie w o f wha t i s alread y understoo d i n Chapters 2 and 3 .

IX

x FOREWOR D

Schottky uniformizatio n provide s a visualizatio n o f Arakelov' s ge -ometry a t arithmeti c infinity , whic h serve s a s th e mai n motivatio n o f Chapter 4 .

Among th e mos t tantalizin g development s i s th e recurren t emer -gence of patches o f common ground fo r numbe r theor y an d theoretica l physics.

In fact , on e can presen t i n thi s ligh t th e famou s theore m o f youn g Gauss characterisin g regula r polygon s tha t ca n b e constructe d usin g only rule r an d compass . I n hi s Tagebuch entry o f Marc h 3 0 h e an -nounced tha t a regular 17-go n has thi s property .

Somewhat modernizin g hi s discovery , on e can presen t i t i n th e fol -lowing way.

In th e comple x plane , root s o f unit y o f degre e n for m vertice s o f a regula r n-gone . Henc e i t make s sens e t o imagin e tha t w e stud y th e ruler and compass constructions as well not in the Euclidean, but i n the complex plane. Thi s has an unexpected consequence: w e can character -ize the se t o f all points constructibl e i n this way as the maximal Galoi s 2-extension of Q. I t remains to calculate the Galoi s group of Q(e 2m^17): since i t i s cyclic o f orde r 16 , this roo t o f unity i s constructible . More -over, th e sam e i s tru e fo r al l p-gon s wher e p i s a prim e o f th e for m 2n + 1 but no t fo r othe r primes .

A remarkabl e featur e o f thi s resul t i s th e appearanc e o f a hidde n symmetry group , I n fact , th e definition s o f a regula r n-go n an d rule r and compas s construction s ar e initiall y formulate d i n term s o f Eu -clidean plan e geometr y an d sugges t tha t th e relevan t symmetr y grou p must b e tha t o f rigi d rotation s SO (2) , eventuall y extende d b y reflec -tions and shifts . Thi s conclusion turns out t o be totally misleading: in -stead, on e should rel y upon Ga l (Q/Q). Th e actio n of the latte r grou p upon root s o f unit y o f degre e n factor s throug h th e maxima l abelia n quotient an d i s give n b y £ H- > ( k

} wit h k runnin g ove r al l k mo d n with (kjU) = 1 , wherea s th e actio n o f th e rotatio n grou p i s give n b y C ^ Co C wi th C o running ove r al l n-t h roots . Thus , th e Gal(Q/Q) -symmetry doe s not conserv e angle s between vertice s which seem to b e basic fo r th e initia l problem . Instead , i t i s compatibl e wit h additio n and multiplication o f complex numbers, an d this property prove s to b e crucial.

With som e stretc h o f imagination , on e ca n recogniz e i n th e Eu -clidean avata r o f thi s pictur e a physic s flavor (puttin g i t somewha t pompously, i t appeal s t o th e kinematic s o f 2-dimensiona l rigi d bodie s

FOREWORD x i

in gravitationa l vacuum) , wherea s the Galoi s avata r definitel y belong s to numbe r theory .

In th e Marcolli lectures , stressin g numbe r theory , physic s theme s pop u p a t th e en d o f Chapte r 2 (Chaoti c Cosmolog y i n genera l rela -tivity), th e beginnin g o f Chapte r 3 (formalis m o f quantu m statistica l mechanics), and finally, sec. 5 of Chapter 4 where some models of black holes in genera l relativity tur n ou t t o hav e the sam e mathematica l de -scription a s oo-adic fibers of curves in Arakelov geometry. Th e reemer -gence o f Gauss ' Galoi s grou p Gal a6 (Q/Q) i n Bost-Conne s symmetr y breaking, an d o f Gauss ' statistic s o f continued fraction s i n the Chaoti c Cosmology models , shows that connection s wit h classica l mathematic s are a s strong a s ever .

Hopefully, thi s livel y expositio n wil l attrac t youn g researcher s an d incite the m t o engag e themselve s i n exploratio n o f th e ric h ne w terri -tory.

Yuri I. Manin. Bonn, March 17, 2005.

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Titles i n Thi s Serie s

36 Mati ld e Marcolli , Arithmeti c noncommutativ e geometry , 200 5

35 Luc a Capogna , Carlo s E . Kenig , an d Loredan a Lanzani , Harmoni c measure : Geometric an d analyti c point s o f view , 200 5

34 E . B . Dynkin , Superdiffusion s an d positiv e solution s o f nonlinea r partia l differentia l equations, 200 4

33 Kristia n Seip , Interpolatio n an d samplin g i n space s o f analyti c functions , 200 4

32 Pau l B . Larson , Th e stationar y tower : Note s o n a cours e b y W . Hug h Woodin , 200 4

31 Joh n Roe , Lecture s o n coars e geometry , 200 3

30 Anato l e Katok , Combinatoria l construction s i n ergodi c theor y an d dynamics , 200 3

29 Thoma s H . Wolf f (Izabell a Lab a an d Caro l Shubin , editors) , Lecture s o n harmoni c

analysis, 200 3

28 Ski p Garibaldi , Alexande r Merkurjev , an d Jean-Pierr e Serre , Cohomologica l invariants i n Galoi s cohomology , 200 3

27 Sun-Yun g A . Chang , Pau l C . Yang , Karste n Grove , an d Jo n G . Wolfson , Conformal, Riemannia n an d Lagrangia n geometry , Th e 200 0 Barret t Lectures , 200 2

26 Susum u Ariki , Representation s o f quantu m algebra s an d combinatoric s o f Youn g tableaux, 200 2

25 Wil l ia m T . Ros s an d Harol d S . Shapiro , Generalize d analyti c continuation , 200 2

24 Victo r M . Buchstabe r an d Tara s E . Panov , Toru s action s an d thei r application s i n topology an d combinatorics , 200 2

23 Lui s Barreir a an d Yako v B . Pesin , Lyapuno v exponent s an d smoot h ergodi c theory , 2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations , 2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d modular functors , 200 1

20 Al iso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f point s o n surfaces , 199 9

17 M a r c e l B e r g e r , Riemannia n geometr y durin g th e secon d hal f o f th e twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Probeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r K a c , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d S tu rmfe l s , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf F an d Die tma r Salamon , J-holomorphi c curve s an d quantu m cohomology , 1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra , 1993

TITLES I N THI S SERIE S

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o low-dimensional topology , 198 9