Random Matrices, Frobenius Eigenvalues, and Monodromy

Random Matrices, Frobenius Eigenvalues, and Monodromy

by Nicholas M. Katz, Peter Sarnak
     
 

ISBN-10: 0821810170

ISBN-13: 9780821810170

Pub. Date: 09/01/1998

Publisher: American Mathematical Society

The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts

Overview

The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.

Product Details

ISBN-13:
9780821810170
Publisher:
American Mathematical Society
Publication date:
09/01/1998
Series:
Colloquium Publications Series, #45
Pages:
419
Product dimensions:
7.09(w) x 10.63(h) x (d)

Table of Contents

Introduction1
Chapter 1.Statements of the Main Results17
1.0.Measures attached to spacings of eigenvalues17
1.1.Expected values of spacing measures23
1.2.Existence, universality and discrepancy theorems for limits of expected values of spacing measures: the three main theorems24
1.3.Interlude: A functorial property of Haar measure on compact groups25
1.4.Application: Slight economies in proving Theorems 1.2.3 and 1.2.625
1.5.Application: An extension of Theorem 1.2.626
1.6.Corollaries of Theorem 1.5.328
1.7.Another generalization of Theorem 1.2.630
1.8. AppendixContinuity properties of "the i'th eigenvalue" as a function on U(N)32
Chapter 2.Reformulation of the Main Results35
2.0."Naive" versions of the spacing measures35
2.1.Existence, universality and discrepancy theorems for limits of expected values of naive spacing measures: the main theorems bis37
2.2.Deduction of Theorems 1.2.1, 1.2.3 and 1.2.6 from their bis versions38
2.3.The combinatorics of spacings of finitely many points on a line: first discussion42
2.4.The combinatorics of spacings of finitely many points on a line: second discussion45
2.5.The combinatorics of spacings of finitely many points on a line: third discussion: variations on Sep(a) and Clump(a)49
2.6.The combinatiorics of spacings of finitely many points of a line: fourth discussion: another variation on Clump(a)54
2.7.Relation to naive spacing measures on G(N): Int, Cor and TCor54
2.8.Expected value measures via INT and COR and TCOR57
2.9.The axiomatics of proving Theorem 2.1.358
2.10.Large N COR limits and formulas for limit measures63
2.11. AppendixDirect image properties of the spacing measures65
Chapter 3.Reduction Steps in Proving the Main Theorems73
3.0.The axiomatics of proving Theorems 2.1.3 and 2.1.573
3.1.A mild generalization of Theorem 2.1.5: the [open phi]-version74
3.2.M-grid discrepancy, L cutoff and dependence on the choice of coordinates77
3.3.A weak form of Theorem 3.1.689
3.4.Conclusion of the axiomatic proof of Theorem 3.1.690
3.5.Making explicit the constants98
Chapter 4.Test Functions101
4.0.The classes T(n) and T[subscript 0](n) of test functions101
4.1.The random variable Z[n, F, G(N)] on G(N) attached to a function F in T(n)103
4.2.Estimates for the expectation E(Z[n, F, G(N)]) and variance Var(Z[n, F, G(N)]) of Z[n, F, G(N)] on G(N)104
Chapter 5.Haar Measure107
5.0.The Weyl integration formula for the various G(N)107
5.1.The K[subscript N](x, y) version of the Weyl integration formula109
5.2.The L[subscript N](x, y) rewriting of the Weyl integration formula116
5.3.Estimates for L[subscript N](x, y)117
5.4.The L[subscript N](x,y) determinants in terms of the sine ratios S[subscript N](x)118
5.5.Case by case summary of explicit Weyl measure formulas via S[subscript N]120
5.6.Unified summary of explicit Weyl measure formulas via S[subscript N]121
5.7.Formulas for the expectation E(Z[n, F, G(N)])122
5.8.Upper bound for E(Z[n, F, G(N)])123
5.9.Interlude: The sin([pi]x)/[pi]x kernel and its approximations124
5.10.Large N limit of E(Z[n, F, G(N)]) via the sin ([pi]x)/[pi]x kernel127
5.11.Upper bound for the variance133
Chapter 6.Tail Estimates141
6.0.Review: Operators of finite rank and their (reversed) characteristic polynomials141
6.1.Integral operators of finite rank: a basic compatibility between spectral and Fredholm determinants141
6.2.An integration formula143
6.3.Integrals of determinants over G(N) as Fredholm determinants145
6.4.A new special case: O_(2N + 1)151
6.5.Interlude: A determinant-trace inequality154
6.6.First application of the determinant-trace inequality156
6.7.Application: Estimates for the numbers eigen(n, s, G(N))159
6.8.Some curious identities among various eigen(n, s, G(N))162
6.9.Normalized "n'th eigenvalue" measures attached to G(N)163
6.10.Interlude: Sharper upper bounds for eigen(0, s, SO(2N)), for eigen(0, s, O_(2N + 1)), and for eigen(0, s, U(N))166
6.11.A more symmetric construction of the "n'th eigenvalue" measures [nu](n, U(N))169
6.12.Relation between the "n'th eigenvalue" measures [nu](n, U(N)) and the expected value spacing measures [mu](U(N), sep. k) on a fixed U(N)170
6.13.Tail estimate for [mu](U(N), sep. 0) and [mu](univ, sep. 0)174
6.14.Multi-eigenvalue location measures, static spacing measures and expected values of several variable spacing measures on U(N)175
6.15.A failure of symmetry183
6.16.Offset spacing measures and their relation to multi-eigenvalue location measures on U(N)185
6.17.Interlude: "Tails" of measures on R[superscript r]189
6.18.Tails of offset spacing measures and tails of multi-eigenvalue location measures on U(N)192
6.19.Moments of offset spacing measures and of multi-eigenvalue location measures on U(N)194
6.20.Multi-eigenvalue location measures for the other G(N)195
Chapter 7.Large N Limits and Fredholm Determinants197
7.0.Generating series for the limit measures [mu](univ, spe.'s a) in several variables: absolute continuity of these measures197
7.1.Interlude: Proof of Theorem 1.7.6205
7.2.Generating series in the case r = 1: relation to a Fredholm determinant208
7.3.The Fredholm determinants E(T, s) and E[plus or minus](T, s)211
7.4.Interpretation of E(T, s) and E[plus or minus](T, s) as large N scaling limits of E(N, T, s) and E[plus or minus](N, T, s)212
7.5.Large N limits of the measures [nu](n, G(N)): the measures [nu](n) and [nu]([plus or minus], n)215
7.6.Relations among the measures [mu][subscript n] and the measures [nu](n)225
7.7.Recapitulation, and concordance with the formulas in [Mehta]228
7.8.Supplement: Fredholm determinants and spectral determinants, with applications to E(T, s) and E[plus or minus](T, s)229
7.9.Interlude: Generalities on Fredholm determinants and spectral determinants232
7.10.Application to E(T, s) and E[plus or minus](T, s)235
7.11. AppendixLarge N limits of multi-eigenvalue location measures and of static and offset spacing measures on U(N)235
Chapter 8.Several Variables245
8.0.Fredholm determinants in several variables and their measure-theoretic meaning (cf. [T-W])245
8.1.Measure-theoretic application to the G(N)248
8.2.Several variable Fredholm determinants for the sin ([pi]x)/[pi]x kernel and its [plus or minus] variants249
8.3.Large N scaling limits251
8.4.Large N limits of multi-eigenvalue location measures attached to G(N)257
8.5.Relation of the limit measure Off [mu](univ, offsets c) with the limit measures [nu](c)263
Chapter 9.Equidistribution267
9.0.Preliminaries267
9.1.Interlude: zeta functions in families: how lisse pure F's arise in nature270
9.2.A version of Deligne's equidistribution theorem275
9.3.A uniform version of Theorem 9.2.6279
9.4.Interlude: Pathologies around (9.3.7.1)280
9.5.Interpretation of (9.3.7.2)283
9.6.Return to a uniform version of Theorem 9.2.6283
9.7.Another version of Deligne's equidistribution theorem287
Chapter 10.Monodromy of Families of Curves293
10.0.Explicit families of curves with big G[subscript geom]293
10.1.Examples in odd characteristic293
10.2.Examples in characteristic two301
10.3.Other examples in odd characteristic302
10.4.Effective constants in our examples303
10.5.Universal families of curves of genus g [greater than or equal] 2304
10.6.The moduli space M[subscript g,3K] for g [greater than or equal] 2307
10.7.Naive and intrinsic measures on U Sp(2g) # attached to universal families of curves315
10.8.Measures on U Sp(2g) # attached to universal families of hyperelliptic curves320
Chapter 11.Monodromy of Some Other Families323
11.0.Universal families of principally polarized abelian varieties323
11.1.Other "rational over the base field" ways of rigidifying curves and abelian varieties324
11.2.Automorphisms of polarized abelian varieties327
11.3.Naive and intrinsic measures on U Sp(2g) # attached to universal families of principally polarized abelian varieties328
11.4.Monodromy of universal families of hypersurfaces331
11.5.Projective automorphisms of hypersurfaces335
11.6.First proof of 11.5.2335
11.7.Second proof of 11.5.2337
11.8.A properness result342
11.9.Naive and intrinsic measures on U Sp(prim(n,d)) # (if n is odd) or on O(prim(n, d)) # (if n is even) attached to universal families of smooth hypersurfaces of degree d in P[superscript n+1]346
11.10.Monodromy of families of Kloosterman sums347
Chapter 12.GUE Discrepancies in Various Families351
12.0.A basic consequence of equidistribution: axiomatics351
12.1.Application to GUE discrepancies352
12.2.GUE discrepancies in universal families of curves353
12.3.GUE discrepancies in universal families of abelian varieties355
12.4.GUE discrepancies in universal families of hypersurfaces356
12.5.GUE discrepancies in families of Kloosterman sums358
Chapter 13.Distribution of Low-lying Frobenius Eigenvalues in Various Families361
13.0.An elementary consequence of equidistribution361
13.1.Review of the measures [nu](c, G(N))363
13.2.Equidistribution of low-lying eigenvalues in families of curves according to the measure [nu](c, U Sp(2g))364
13.3.Equidistribution of low-lying eigenvalues in families of abelian varieties according to the measure [nu](c, U Sp(2g))365
13.4.Equidistribution of low-lying eigenvalues in families of odd-dimensional hypersurfaces according to the measure [nu](c, U Sp(prim(n,d)))366
13.5.Equidistribution of low-lying eigenvalues of Kloosterman sums in evenly many variables according to the measure [nu](c, U Sp(2n))367
13.6.Equidistribution of low-lying eigenvalues of characteristic two Kloosterman sums in oddly many variables according to the measure [nu](c, SO(2n+1))367
13.7.Equidistribution of low-lying eigenvalues in families of even-dimensional hypersurfaces according to the measures [nu](c, SO(prim(n,d))) and [nu](c,O_(prim(n,d)))368
13.8.Passage to the large N limit369
AppendixDensities373
AD.0.Overview373
AD.1.Basic definitions: W[subscript n](f, A, G(N)) and W[subscript n](f, G(N))373
AD.2.Large N limits: the easy case374
AD.3.Relations between eigenvalue location measures and densities: generalities378
AD.4.Second construction of the large N limits of the eigenvalue location measures [nu](c, G(N)) for G(N) one of U(N), SO(2N + 1), U Sp(2N), SO(2N), O_(2N + 2), O_(2N + 1)381
AD.5.Large N limits for the groups U[subscript k](N): Widom's result385
AD.6.Interlude: The quantities V[subscript r]([open phi], U[subscript k](N)) and V[subscript r]([open phi], U (N))386
AD.7.Interlude: Integration formulas on U (N) and on U[subscript k](N)390
AD.8.Return to the proof of Widom's theorem392
AD.9.End of the proof of Theorem AD.5.2399
AD.10.Large N limits of the eigenvalue location measures on the U[subscript k](N)401
AD.11.Computation of the measures v(c) via low-lying eigenvalues of Kloosterman sums in oddly many variables in odd characteristic403
AD.12.A variant of the one-level scaling density405
AppendixGraphs411
AG.0.How the graphs were drawn, and what they show411
References417

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