Random Matrices, Frobenius Eigenvalues, and Monodromy 1470431912, 9781470431914

Table of contents :
Cover
Title page
Contents
Introduction
Statements of the main results
Reformulation of the main results
Reduction steps in proving the main theorems
Test functions
Haar measure
Tail estimates
Large N limits and Fredholm determinants
Several variables
Equidistribution
Monodromy of families of curves
Monodromy of some other families
GUE discrepancies in various families
Distribution of low-lying Frobenius eigenvalues in various families
Appendix: Densities
Appendix: Graphs
References
Back Cover

Citation preview

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Selected Title s i n Thi s Serie s 45 Nichola s M . Kat z an d P e t e r Sarnak , Rando m matrices , Frobeniu s eigenvalues , an d monodromy, 199 9 44 M a x - A l b e r t Knus , A l e x a n d e r Merkurjev , an d M a r k u s R o s t , Th e boo k o f involutions, 199 8 43 Lui s A . Caffarell i an d Xavie r Cabre , Full y nonlinea r ellipti c equations , 199 5 42 Victo r Guillemi n an d S h l o m o Sternberg , Variation s o n a them e b y Kepler , 199 0 41 Alfre d Tarsk i an d S t e v e n Givant , A formalizatio n o f se t theor y withou t variables , 198 7 40 R . H . Bing , Th e geometri c topolog y o f 3-manifolds , 198 3 39 N . J a c o b s o n , Structur e an d representation s o f Jorda n algebras , 196 8 38 O . Ore , Theor y o f graphs , 196 2 37 N . J a c o b s o n , Structur e o f rings , 195 6 36 W . H . Gottschal k an d G . A . H e d l u n d , Topologica l dynamics , 195 5 35 A . C . Schaeffe r an d D . C . Spencer , Coefficien t region s fo r Schlich t functions , 195 0 34 J . L . Walsh , Th e locatio n o f critica l point s o f analyti c an d harmoni c functions , 195 0 33 J . F . R i t t , Differentia l algebra , 195 0 32 R . L . W i l d e r , Topolog y o f manifolds , 194 9 31 E . Hill e a n d R . S . Phillips , Functiona l analysi s an d semigroups , 195 7 30 T . R a d o , Lengt h an d area , 194 8 29 A . Weil , Foundation s o f algebrai c geometry , 194 6 28 G . T . W h y b u r n , Analyti c topology , 194 2 27 S . Lefschetz , Algebrai c topology , 194 2 26 N . Levinson , Ga p an d densit y theorems , 194 0 25 Garret t Birkhoff , Lattic e theory , 194 0 24 A . A . A l b e r t , Structur e o f algebras , 193 9 23 G . Szego , Orthogona l polynomials , 193 9 22 C . N . M o o r e , Summabl e serie s an d convergenc e factors , 193 8 21 J . M . T h o m a s , Differentia l systems , 193 7 20 J . L . Walsh , Interpolatio n an d approximatio n b y rationa l function s i n th e comple x domain, 193 5 19 R . E . A . C . P a l e y an d N . W i e n e r , Fourie r transform s i n th e comple x domain , 193 4 18 M . Morse , Th e calculu s o f variation s i n th e large , 193 4 17 J . M . W e d d e r b u r n , Lecture s o n matrices , 193 4 16 G . A . Bliss , Algebrai c functions , 193 3 15 M . H . S t o n e , Linea r transformation s i n Hilber t spac e an d thei r application s t o analysis , 1932 14 J . F . R i t t , Differentia l equation s fro m th e algebrai c standpoint , 193 2 13 R . L . M o o r e , Foundation s o f poin t se t theory , 193 2 12 S . Lefschetz , Topology , 193 0 11 D . Jackson , Th e theor y o f approximation , 193 0 10 A . B . C o b l e , Algebrai c geometr y an d thet a functions , 192 9 9 G . D . Birkhoff , Dynamica l systems , 192 7 8 L . P . Eisenhart , Non-Riemannia n geometry , 192 7 7 E . T . Bell , Algebrai c arithmetic , 192 7 6 G . C . E v a n s , Th e logarithmi c potential , discontinuou s Dirichle t an d Neuman n problems , 1927

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Rando m Matrices , Frobeniu s Eigenvalues , an d Monodrom y

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http://dx.doi.org/10.1090/coll/045

America n Mathematica l Societ y Colloquiu m Publication s Volum e 45

Rando m Matrices , Frobeniu s Eigenvalues , an d Monodrom y Nichola s M. Kat z Pete r Sarna k

America n Mathematica l Societ y Providence , Rhod e Islan d Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

Editorial Boar d Joan S . Birma n Susan J . Friedlander , Chai r Stephen Lichtenbau m

1991 Mathematics Subject

Classification. P r i m a r y 11G25 , 14G10 , 60Fxx , 14D05 ; Secondary 11M06 , 82Bxx , 11Y35 .

ABSTRACT. Th e mai n topi c o f thi s boo k i s th e dee p relatio n betwee n th e spacing s betwee n zero s of zet a an d L-function s an d spacing s betwee n eigenvalue s o f rando m element s o f larg e compac t classical groups . Thi s relation , th e Montgomery-Odlyzk o law , i s show n t o hol d fo r wid e classe s o f zeta an d L-function s ove r finite fields. The boo k draw s on , an d give s accessibl e account s of , man y disparat e area s o f mathematics , from algebrai c geometry , modul i spaces , monodromy , equidistribution , an d th e Wei l Conjectures , to probabilit y theor y o n th e compac t classica l group s i n th e limi t a s thei r dimensio n goe s t o infinity an d relate d technique s fro m orthogona l polynomial s an d Fredhol m determinants . I t wil l be usefu l an d interestin g t o researcher s an d graduat e student s workin g i n an y o f thes e areas .

Library o f Congres s Cataloging-in-Publicatio n D a t a Katz, Nichola s M. , 1943 Random matrices , Probeniu s eigenvalues , an d monodrom y / Nichola s M . Katz , Pete r Sarnak . p. cm . — (Colloquiu m publication s / America n Mathematica l Society , ISS N 0065-925 8 ; v. 45 ) Includes bibliographica l references . ISBN 0-8218-1017- 0 (hardcove r : alk . paper ) 1. Functions , Zeta . 2 . L-functions . 3 . Rando m matrices . 4 . Limi t theorem s (Probabilit y theory) 5 . Monodrom y groups . I . Sarnak , Peter . II . Title . III . Series : Colloquiu m publica tions (America n Mathematica l Society ) ; v. 45 . QA351.K36 199 8 515'.56—dc21 98-2045 9 CIP

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o [email protected] . © 199 9 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0

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Contents Introduction 1 Chapter 1 . Statement s o f th e Mai n Result s 1 1.0. Measure s attache d t o spacing s o f eigenvalue s 1 1.1. Expecte d value s o f spacin g measure s 2 1.2. Existence , universalit y an d discrepanc y theorem s fo r limit s o f expected value s o f spacin g measures : th e thre e mai n theorem s 2 1.3. Interlude : A functorial propert y o f Haa r measur e o n compac t group s 2 1.4. Application : Sligh t economie s i n provin g Theorem s 1.2. 3 an d 1.2. 6 2 1.5. Application : A n extensio n o f Theore m 1.2. 6 2 1.6. Corollarie s o f Theore m 1.5. 3 2 1.7. Anothe r generalizatio n o f Theore m 1.2. 6 3 1.8. Appendix : Continuit y propertie s o f "th e i't h eigenvalue " a s a function o n U(N) 3 Chapter 2 . Reformulatio n o f th e Mai n Result s 3 2.0. "Naive " version s o f th e spacin g measure s 3 2.1. Existence , universalit y an d discrepanc y theorem s fo r limit s o f expected value s o f naiv e spacin g measures : th e mai n theorem s bi s 3 2.2. Deductio n o f Theorem s 1.2.1 , 1.2. 3 an d 1.2. 6 fro m thei r bi s version s 3 2.3. Th e combinatoric s o f spacing s o f finitel y man y point s o n a line : first discussio n 4 2.4. Th e combinatoric s o f spacing s o f finitel y man y point s o n a line : second discussio n 4 2.5. Th e combinatoric s o f spacing s o f finitel y man y point s o n a line : third discussion : variation s o n Sep(a ) an d Clum p (a) 4 2.6. Th e combinatoric s o f spacing s o f finitel y man y point s o f a line : fourth discussion : anothe r variatio n o n Clum p (a) 5 2.7. Relatio n t o naiv e spacin g measure s o n G(N): Int , Co r an d TCo r 5 2.8. Expecte d valu e measure s vi a IN T an d CO R an d TCO R 5 2.9. Th e axiomatic s o f provin g Theore m 2.1. 3 5 2.10. Larg e N CO R limit s an d formula s fo r limi t measure s 6 2.11. Appendix : Direc t imag e propertie s o f th e spacin g measure s 6 Chapter 3 . Reductio n Step s i n Provin g th e Mai n Theorem s 7 3.0. Th e axiomatic s o f provin g Theorem s 2.1. 3 an d 2.1. 5 7 3.1. A mil d generalizatio n o f Theore m 2.1.5 : th e y>versio n 7 3.2. M-gri d discrepancy , L cutof f an d dependenc e o n th e choic e o f coordinates 7 3.3. A wea k for m o f Theore m 3.1. 6 8

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3.4. Conclusio n o f th e axiomati c proo f o f Theore m 3.1. 6 9 3.5. Makin g explici t th e constant s 9 Chapter 4 . Tes t Function s 10 4.0. Th e classe s T(n) an d 7o(n ) o f tes t function s 10 4.1. Th e rando m variabl e Z[n,F,G(N)] o n G(N) attache d t o a function F i n T(n) 10 4.2. Estimate s fo r th e expectatio n E(Z[n,F,G(N)]) an d varianc e Var(Z[n, F, G(N)}) o f Z[n , F, G(N)] o n G(N) 10

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Chapter 5 . Haa r Measur e 10 5.0. Th e Wey l integratio n formul a fo r th e variou s G(N) 10 5.1. Th e KN(x,y) versio n o f th e Wey l integratio n formul a 10 5.2. Th e Ljsi{x,y) rewritin g o f th e Wey l integratio n formul a 11 5.3. Estimate s fo r LN(X, y) 111 5.4. Th e L,N(x,y) determinant s i n term s o f th e sin e ratio s SN(X) 11 5.5. Cas e b y cas e summar y o f explici t Wey l measur e formula s vi a SN 12 5.6. Unifie d summar y o f explici t Wey l measur e formula s vi a SN 12 5.7. Formula s fo r th e expectatio n E(Z[n, F, G(N)]) 12 5.8. Uppe r boun d fo r E(Z[n, F, G(N)]) 12 5.9. Interlude : Th e sin(7ra;)/7r £ kerne l an d it s approximation s 12 5.10. Larg e N limi t o f E(Z[n, F, G(N)]) vi a th e sin(7nr)/7rx kerne l 12 5.11. Uppe r boun d fo r th e varianc e 13

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Chapter 6 . Tai l Estimate s 14 6.0. Review : Operator s o f finit e ran k an d thei r (reversed ) characteristi c polynomials 14 6.1. Integra l operator s o f finit e rank : a basi c compatibilit y betwee n spectral an d Fredhol m determinant s 14 6.2. A n integratio n formul a 14 6.3. Integral s o f determinant s ove r G(N) a s Fredhol m determinant s 14 6.4. A ne w specia l case : 0_(2i V + 1 ) 15 6.5. Interlude : A determinant-trac e inequalit y 15 6.6. Firs t applicatio n o f th e determinant-trac e inequalit y 15 6.7. Application : Estimate s fo r th e number s eigen(n , s, G(N)) 15 6.8. Som e curiou s identitie s amon g variou s eigen(n , s, G(iV)) 16 6.9. Normalize d "n't h eigenvalue " measure s attache d t o G(N) 16 6.10. Interlude : Sharpe r uppe r bound s fo r eigen(0 , s,SO(2N)), fo r eigen(0, s, 0-(2N + 1)), an d fo r eigen(0,s , U(N)) 16 6.11. A mor e symmetri c constructio n o f th e "n't h eigenvalue " measures v(n,U{N)) 16 6.12. Relatio n betwee n th e "n't h eigenvalue " measure s v(n,U{N)) and th e expecte d valu e spacin g measure s /J,(U(N), sep . fc) on a fixe d U(N) 17 6.13. Tai l estimat e fo r IM(U(N), sep . 0 ) an d /x(univ , sep . 0 ) 17 6.14. Multi-eigenvalu e locatio n measures , stati c spacin g measure s an d expected value s o f severa l variabl e spacin g measure s o n U(N) 17 6.15. A failur e o f symmetr y 18 6.16. Offse t spacin g measure s an d thei r relatio n t o multi-eigenvalu e location measure s o n U(N) 18

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6.17. Interlude : "Tails " o f measure s o n W 18 6.18. Tail s o f offse t spacin g measure s an d tail s o f multi-eigenvalu e location measure s o n U(N) 19 6.19. Moment s o f offse t spacin g measure s an d o f multi-eigenvalu e location measure s o n U(N) 19 6.20. Multi-eigenvalu e locatio n measure s fo r th e othe r G(N) 19

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Chapter 7 . Larg e i V Limits an d Predhol m Determinant s 19 7.0. Generatin g serie s fo r th e limi t measure s /i(univ , sep.' s a) i n severa l variables: absolut e continuit y o f thes e measure s 19 7.1. Interlude : Proo f o f Theore m 1.7. 6 20 7.2. Generatin g serie s i n th e cas e r = 1 : relatio n t o a Predhol m determinant 20 7.3. Th e Predhol m determinant s E{T, s) an d E±(T, s) 21 7.4. Interpretatio n o f E(T,s) an d E±(T,s) a s larg eA T scaling limit s of E(N, T , s) an d E± (AT, T, s) 21 7.5. Larg eT V limits o f th e measure s v(n,G(N)): th e measure s i/(n) and v(±,n) 21 7.6. Relation s amon g th e measure s \x n an d th e measure s v{n) 22 7.7. Recapitulation , an d concordanc e wit h th e formula s i n [Mehta ] 22 7.8. Supplement : Predhol m determinant s an d spectra l determinants , with application s t o E(T, s) an d E±(T, s) 22 7.9. Interlude : Generalitie s o n Predhol m determinant s an d spectra l determinants 23 7.10. Applicatio n t o E(T, s) an d E±(T, s) 23 7.11. Appendix : Larg e N limit s o f multi-eigenvalu e locatio n measure s and o f stati c an d offse t spacin g measure s o n U(N) 23

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Chapter 8 . Severa l Variable s 24 8.0. Predhol m determinant s i n severa l variable s an d thei r measure theoretic meanin g (cf . [T-W] ) 24 8.1. Measure-theoreti c applicatio n t o th e G(N) 24 8.2. Severa l variabl e Predhol m determinant s fo r th e sin(7rx)/7r x kerne l and it s ± variant s 24 8.3. Larg e N scalin g limit s 25 8.4. Larg eA T limits o f multi-eigenvalu e locatio n measure s attache d t o G(N) 25 8.5. Relatio n o f th e limi t measur e Off/x(univ , offset s c ) wit h th e limi t measures u(c) 26

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Chapter 9 . Equidistributio n 26 9.0. Preliminarie s 26 9.1. Interlude : zet a function s i n families : ho w liss e pur e .P s aris e i n nature 27 9.2. A version o f Deligne' s equidistributio n theore m 27 9.3. A unifor m versio n o f Theore m 9.2. 6 27 9.4. Interlude : Pathologie s aroun d (9.3.7.1 ) 28 9.5. Interpretatio n o f (9.3.7.2 ) 28 9.6. Retur n t o a unifor m versio n o f Theore m 9.2. 6 28 9.7. Anothe r versio n o f Deligne' s equidistributio n theore m 28

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Chapter 10 . Monodrom y o f Familie s o f Curve s 29 10.0. Explici t familie s o f curve s wit h bi g G ge0m 29 10.1. Example s i n od d characteristi c 29 10.2. Example s i n characteristi c tw o 30 10.3. Othe r example s i n od d characteristi c 30 10.4. Effectiv e constant s i n ou r example s 30 10.5. Universa l familie s o f curve s o f genu s g > 2 30 10.6. Th e modul i spac e M 9^K ^v g >2 30 10.7. Naiv e an d intrinsi c measure s o n USp(2g)# attache d t o universa l families o f curve s 31 10.8. Measure s o n USp(2g)# attache d t o universa l familie s o f hyperelliptic curve s 32

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Chapter 11 . Monodrom y o f Som e Othe r Familie s 32 11.0. Universa l familie s o f principall y polarize d abelia n varietie s 32 11.1. Othe r "rationa l ove r th e bas e field" way s o f rigidifyin g curve s and abelia n varietie s 32 11.2. Automorphism s o f polarized abelia n varietie s 32 11.3. Naiv e an d intrinsi c measure s o n USp(2g)# attache d t o universal familie s o f principall y polarize d abelia n varietie s 32 11.4. Monodrom y o f universa l familie s o f hypersurface s 33 11.5. Projectiv e automorphism s o f hypersurface s 33 11.6. Firs t proo f o f 11.5. 2 33 11.7. Secon d proo f o f 11.5. 2 33 11.8. A propernes s resul t 34 11.9. Naiv e an d intrinsi c measure s o n U Sp(prim(n, d))# (i f n i s odd ) or o n 0(prim(n,d)) # (i f n i s even ) attache d t o universa l families o f smoot h hypersurface s o f degre e d i n P n + 1 34 11.10. Monodrom y o f familie s o f Kloosterma n sum s 34

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Chapter 12 . GU E Discrepancie s i n Variou s Familie s 35 12.0. A basi c consequenc e o f equidistribution : axiomatic s 35 12.1. Applicatio n t o GU E discrepancie s 35 12.2. GU E discrepancie s i n universa l familie s o f curve s 35 12.3. GU E discrepancie s i n universa l familie s o f abelia n varietie s 35 12.4. GU E discrepancie s i n universa l familie s o f hypersurface s 35 12.5. GU E discrepancie s i n familie s o f Kloosterma n sum s 35 Chapter 13 . Distributio n o f Low-lyin g Frobeniu s Eigenvalue s i n Variou s Families 36 13.0. A n elementar y consequenc e o f equidistributio n 36 13.1. Revie w o f th e measure s v(c,G(N)) 36 13.2. Equidistributio n o f low-lyin g eigenvalue s i n familie s o f curve s according t o th e measur e i/(c , USp(2g)) 36 13.3. Equidistributio n o f low-lyin g eigenvalue s i n familie s o f abelia n varieties accordin g t o th e measur e z/(c , USp(2g)) 36 13.4. Equidistributio n o f low-lyin g eigenvalue s i n familie s o f odd dimensional hypersurface s accordin g t o th e measur e i/(c,C/Sp(prim(n,d))) 36 Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

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13.5. Equidistributio n o f low-lyin g eigenvalue s o f Kloosterma n sum s i n evenly man y variable s accordin g t o th e measur e z/(c , USp{2n)) 36 13.6. Equidistributio n o f low-lyin g eigenvalue s o f characteristi c tw o Kloosterman sum s i n oddl y man y variable s accordin g t o th e measure z/(c , SO(2n + 1) ) 36 13.7. Equidistributio n o f low-lyin g eigenvalue s i n familie s o f even dimensional hypersurface s accordin g t o th e measure s i/(c, SO(prim(n, d))) an d z/(c , 0_(prim(n,cQ)) 36 13.8. Passag e t o th e larg e N limi t 36

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Appendix: Densitie s 37 AD.O. Overvie w 37 AD.l. Basi c definitions : W n(f, A, G(N)) an d W n(f, G(N)) 37 AD.2. Larg e N limits : th e eas y cas e 37 AD.3. Relation s betwee n eigenvalu e locatio n measure s an d densities : generalities 37 AD.4. Secon d constructio n o f th e larg e N limit s o f th e eigenvalu e location measure s v(c,G(N)) fo r G(N) on e o f U(N), SO(2N + 1) , USp(2N), SO(2N), 0_(2A r + 2) , 0_(27V + 1 ) 38 AD.5. Larg e N limit s fo r th e group s Uk{N): Widom' s resul t 38 AD.6. Interlude : Th e quantitie s V r((p, U k(N)) an d V r(i) , (9M( 1 , an d conside r th e unitar y grou p U(N) o f size N. Give n a n elemen t A i n U(N), it s N eigenvalue s li e o n th e uni t circle , an d we for m th e N normalize d (t o hav e mea n 1 ) spacing s betwee n pair s o f adjacen t eigenvalues, an d ou t o f thes e N spacing s w e for m th e probabilit y measur e o n R which give s mas s 1/N t o eac h o f th e N normalize d spacings . Thi s measur e w e call [JL(A, U(N)), th e spacin g measur e attache d t o a n elemen t A i n U(N). W e view A— i > /JL(A, U(N)) a s a measure-value d functio n o n U(N). On e ca n mak e sens e o f the integra l o f thi s functio n ove r U(N) agains t th e tota l mas s on e Haa r measur e dA o n U(N): th e resul t make s sens e a s a probabilit y measur e o n R , denote d

»(U(N)):= [

»(A,U(N))dA. JU(N)

One the n show s tha t a sT V grows , th e measure s /J,(U(N)) o n R hav e a limi t whic h is agai n a probabilit y measur e o n R , whic h w e denot e ju(univ) , an d cal l th e GU E measure. 1 On e show s tha t it s cumulativ e distributio n functio n CDF^ ( u n i v ) (x) : = / d/x(univ J [— oo,x ]

)

is continuou s o n R . [I n fact , thi s measur e ha s a density , whic h vanishe s outsid e R>o, an d i s real analyti c o n R>o , cf . Appendix : Graph s fo r a picture. ] For th e applicatio n t o curve s tha t w e hav e i n mind , w e nee d t o kno w tha t w e can obtai n th e GU E measur e no t jus t fro m th e serie s o f unitar y group s U(N), bu t also fro m an y o f th e serie s o f compac t classica l groups . Indeed , suppos e w e ar e given an y compac t subgrou p K o f a give n unitar y grou p U(N). W e can , fo r eac h element Am K, for m th e spacin g measur e attache d t o A though t o f a s a n elemen t of U(N). T o remin d ourselve s tha t w e d o thi s onl y fo r element s o f K, w e denot e this measur e fj,(A,K). The n w e vie w A — i » fj,(A,K) a s a measure-value d functio n on K , an d w e integrat e thi s functio n agains t th e tota l mas s on e Haa r measur e dA ^Tn the physic s literature , thi s measur e ofte n carrie s Wigner' s nam e Licensed to AMS. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms

3

INTRODUCTION

on th e compac t grou p K. Th e result , denote d fJ>(K) := f K fi(A, K) dA, i s itsel f a probability measur e o n R . We can perfor m thi s constructio n wit h K an y o f the compac t classica l groups , U(N) o r SU(N) o r USp{2N) o r SO{2N+l) o r SO{2N) o r 0(2 W + 1 ) or 0(2N) i n their standar d representations . W e sho w tha t fo r G{N) runnin g ove r an y o f thes e series o f compac t classica l groups , th e sequenc e o f probabilit y measure s n{G(N)) on R converges , a s N grows , to th e sam e measur e //(univ) , the GU E measure , tha t we obtaine d a s th e larg e N limi t o f th e fj,(U(N)) measures . [Th e cas e whic h wil l be relevan t t o curve s ove r finite fields wil l tur n ou t t o b e th e compac t symplecti c groups USp(2N).] Now le t u s retur n t o a curv e C/¥ q ove r a finite field, o f som e genu s g. Sinc e the spacin g measur e fi(C/¥ q) give s eac h o f 2g points mas s l/2# , it s CD F i s a ste p function, wit h 2g jumps. S o it canno t possibl y b e the cas e that ji{C/¥ q) i s equal t o the GU E measure , whos e CD F i s continuous. Moreover , a s we shall see later i n thi s Introduction, ove r an y finite field ther e ar e sequence s o f curve s o f increasin g genu s whose spacin g measure s ar e arbitraril y clos e t o th e delt a measur e So supported a t the origin . S o i t i s simpl y no t tru e tha t th e spacin g measure s o f al l curve s o f large genu s ar e clos e t o th e GU E measure . Wha t w e sho w i s tha t "most " curve s of larg e genu s ove r a larg e finite field hav e thei r spacin g measur e quit e clos e to th e GUE measure , o r i n othe r word s tha t "most " curve s o f sufficiently larg e genus ove r a sufficientl y larg e finite field satisf y th e Montgomery-Odlyzk o La w to a n arbitrar y degree o f precision . To mak e thi s mor e precise , w e nee d a numerica l measur e o f ho w clos e tw o probability measure s o n R , sa y fi an d v, are . Fo r this , w e us e th e Kolmogoroff Smirnov discrepancy , define d a s th e greates t vertica l distanc e betwee n th e graph s of thei r CDF's : discrep(/x, v) : = Su p | CDFM (s) - CDF„(s)| . sinR

Notice tha t discrep(/^ , v) i s a numbe r whic h alway s lie s in th e close d interva l [0,1] , just becaus e CDF' s o f probabilit y measure s tak e value s i n [0,1] . Now le t u s denot e b y A4 g(¥q) th e set , know n t o b e finite, consistin g o f al l F^-isomorphism classe s o f genu s g curve s ove r ¥ q. Ou r essentia l resul t abou t th e spacing measure s n(C/¥ q) attache d t o curve s ove r finite fields, an d thei r relatio n to th e GU E measur e /i(univ) , i s this: Theorem (cf . 12.2.3) . We have the double limit formula lim li m (l/\MJ¥ q)\) V

discrep(/i(?mw),/j(C/F