# Random Matrices, Frobenius Eigenvalues, and Monodromy

by Nicholas M. Katz, Peter SarnakISBN-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

Introduction | 1 | |

Chapter 1. | Statements of the Main Results | 17 |

1.0. | Measures attached to spacings of eigenvalues | 17 |

1.1. | Expected values of spacing measures | 23 |

1.2. | Existence, universality and discrepancy theorems for limits of expected values of spacing measures: the three main theorems | 24 |

1.3. | Interlude: A functorial property of Haar measure on compact groups | 25 |

1.4. | Application: Slight economies in proving Theorems 1.2.3 and 1.2.6 | 25 |

1.5. | Application: An extension of Theorem 1.2.6 | 26 |

1.6. | Corollaries of Theorem 1.5.3 | 28 |

1.7. | Another generalization of Theorem 1.2.6 | 30 |

1.8. Appendix | Continuity properties of "the i'th eigenvalue" as a function on U(N) | 32 |

Chapter 2. | Reformulation of the Main Results | 35 |

2.0. | "Naive" versions of the spacing measures | 35 |

2.1. | Existence, universality and discrepancy theorems for limits of expected values of naive spacing measures: the main theorems bis | 37 |

2.2. | Deduction of Theorems 1.2.1, 1.2.3 and 1.2.6 from their bis versions | 38 |

2.3. | The combinatorics of spacings of finitely many points on a line: first discussion | 42 |

2.4. | The combinatorics of spacings of finitely many points on a line: second discussion | 45 |

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 TCor | 54 |

2.8. | Expected value measures via INT and COR and TCOR | 57 |

2.9. | The axiomatics of proving Theorem 2.1.3 | 58 |

2.10. | Large N COR limits and formulas for limit measures | 63 |

2.11. Appendix | Direct image properties of the spacing measures | 65 |

Chapter 3. | Reduction Steps in Proving the Main Theorems | 73 |

3.0. | The axiomatics of proving Theorems 2.1.3 and 2.1.5 | 73 |

3.1. | A mild generalization of Theorem 2.1.5: the [open phi]-version | 74 |

3.2. | M-grid discrepancy, L cutoff and dependence on the choice of coordinates | 77 |

3.3. | A weak form of Theorem 3.1.6 | 89 |

3.4. | Conclusion of the axiomatic proof of Theorem 3.1.6 | 90 |

3.5. | Making explicit the constants | 98 |

Chapter 4. | Test Functions | 101 |

4.0. | The classes T(n) and T[subscript 0](n) of test functions | 101 |

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 Measure | 107 |

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 formula | 109 |

5.2. | The L[subscript N](x, y) rewriting of the Weyl integration formula | 116 |

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 approximations | 124 |

5.10. | Large N limit of E(Z[n, F, G(N)]) via the sin ([pi]x)/[pi]x kernel | 127 |

5.11. | Upper bound for the variance | 133 |

Chapter 6. | Tail Estimates | 141 |

6.0. | Review: Operators of finite rank and their (reversed) characteristic polynomials | 141 |

6.1. | Integral operators of finite rank: a basic compatibility between spectral and Fredholm determinants | 141 |

6.2. | An integration formula | 143 |

6.3. | Integrals of determinants over G(N) as Fredholm determinants | 145 |

6.4. | A new special case: O_(2N + 1) | 151 |

6.5. | Interlude: A determinant-trace inequality | 154 |

6.6. | First application of the determinant-trace inequality | 156 |

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 symmetry | 183 |

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 Determinants | 197 |

7.0. | Generating series for the limit measures [mu](univ, spe.'s a) in several variables: absolute continuity of these measures | 197 |

7.1. | Interlude: Proof of Theorem 1.7.6 | 205 |

7.2. | Generating series in the case r = 1: relation to a Fredholm determinant | 208 |

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 determinants | 232 |

7.10. | Application to E(T, s) and E[plus or minus](T, s) | 235 |

7.11. Appendix | Large N limits of multi-eigenvalue location measures and of static and offset spacing measures on U(N) | 235 |

Chapter 8. | Several Variables | 245 |

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] variants | 249 |

8.3. | Large N scaling limits | 251 |

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. | Equidistribution | 267 |

9.0. | Preliminaries | 267 |

9.1. | Interlude: zeta functions in families: how lisse pure F's arise in nature | 270 |

9.2. | A version of Deligne's equidistribution theorem | 275 |

9.3. | A uniform version of Theorem 9.2.6 | 279 |

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 9.2.6 | 283 |

9.7. | Another version of Deligne's equidistribution theorem | 287 |

Chapter 10. | Monodromy of Families of Curves | 293 |

10.0. | Explicit families of curves with big G[subscript geom] | 293 |

10.1. | Examples in odd characteristic | 293 |

10.2. | Examples in characteristic two | 301 |

10.3. | Other examples in odd characteristic | 302 |

10.4. | Effective constants in our examples | 303 |

10.5. | Universal families of curves of genus g [greater than or equal] 2 | 304 |

10.6. | The moduli space M[subscript g,3K] for g [greater than or equal] 2 | 307 |

10.7. | Naive and intrinsic measures on U Sp(2g) # attached to universal families of curves | 315 |

10.8. | Measures on U Sp(2g) # attached to universal families of hyperelliptic curves | 320 |

Chapter 11. | Monodromy of Some Other Families | 323 |

11.0. | Universal families of principally polarized abelian varieties | 323 |

11.1. | Other "rational over the base field" ways of rigidifying curves and abelian varieties | 324 |

11.2. | Automorphisms of polarized abelian varieties | 327 |

11.3. | Naive and intrinsic measures on U Sp(2g) # attached to universal families of principally polarized abelian varieties | 328 |

11.4. | Monodromy of universal families of hypersurfaces | 331 |

11.5. | Projective automorphisms of hypersurfaces | 335 |

11.6. | First proof of 11.5.2 | 335 |

11.7. | Second proof of 11.5.2 | 337 |

11.8. | A properness result | 342 |

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 sums | 347 |

Chapter 12. | GUE Discrepancies in Various Families | 351 |

12.0. | A basic consequence of equidistribution: axiomatics | 351 |

12.1. | Application to GUE discrepancies | 352 |

12.2. | GUE discrepancies in universal families of curves | 353 |

12.3. | GUE discrepancies in universal families of abelian varieties | 355 |

12.4. | GUE discrepancies in universal families of hypersurfaces | 356 |

12.5. | GUE discrepancies in families of Kloosterman sums | 358 |

Chapter 13. | Distribution of Low-lying Frobenius Eigenvalues in Various Families | 361 |

13.0. | An elementary consequence of equidistribution | 361 |

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 limit | 369 |

Appendix | Densities | 373 |

AD.0. | Overview | 373 |

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 case | 374 |

AD.3. | Relations between eigenvalue location measures and densities: generalities | 378 |

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 result | 385 |

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 theorem | 392 |

AD.9. | End of the proof of Theorem AD.5.2 | 399 |

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 characteristic | 403 |

AD.12. | A variant of the one-level scaling density | 405 |

Appendix | Graphs | 411 |

AG.0. | How the graphs were drawn, and what they show | 411 |

References | 417 |

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