ISBN-10:
0691123306
ISBN-13:
9780691123301
Pub. Date:
10/02/2005
Publisher:
Princeton University Press
Moments, Monodromy, and Perversity. (AM-159): A Diophantine Perspective. (AM-159)

Moments, Monodromy, and Perversity. (AM-159): A Diophantine Perspective. (AM-159)

by Nicholas M. Katz

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Product Details

ISBN-13: 9780691123301
Publisher: Princeton University Press
Publication date: 10/02/2005
Series: Annals of Mathematics Studies , #173
Pages: 488
Product dimensions: 7.00(w) x 10.00(h) x (d)

About the Author

Nicholas M. Katz is Professor of Mathematics at Princeton University. He is the author of five previous books in this series: Arithmetic Moduli of Elliptic Curves (with Barry Mazur); Gauss Sums, Kloosterman Sums, and Monodromy Groups; Exponential Sums and Differential Equations; Rigid Local Systems; and Twisted L-Functions and Monodromy.

Table of Contents

Introduction1
Chapter 1Basic results on perversity and higher moments9
(1.1)The notion of a d-separating space of functions9
(1.2)Review of semiperversity and perversity12
(1.3)A twisting construction: the object Twist(L,K,F,h)13
(1.4)The basic theorem and its consequences13
(1.5)Review of weights21
(1.6)Remarks on the various notions of mixedness24
(1.7)The Orthogonality Theorem25
(1.8)First Applications of the Orthogonality Theorem31
(1.9)Questions of autoduality: the Frobenius-Schur indicator theorem36
(1.10)Dividing out the "constant part" of an [iota]-pure perverse sheaf42
(1.11)The subsheaf N[subscript ncst0] in the mixed case44
(1.12)Interlude: abstract trace functions; approximate trace functions45
(1.13)Two uniqueness theorems47
(1.14)The central normalization F[subscript 0] of a trace function F50
(1.15)First applications to the objects Twist(L, K, F, h): The notion of standard input52
(1.16)Review of higher moments60
(1.17)Higher moments for geometrically irreducible lisse sheaves61
(1.18)Higher moments for geometrically irreducible perverse sheaves62
(1.19)A fundamental inequality62
(1.20)Higher moment estimates for Twist(L,K,F,h)64
(1.21)Proof of the Higher Moment Theorem 1.20.2: combinatorial preliminaries67
(1.22)Variations on the Higher Moment Theorem76
(1.23)Counterexamples87
Chapter 2How to apply the results of Chapter 193
(2.1)How to apply the Higher Moment Theorem93
(2.2)Larsen's Alternative94
(2.3)Larsen's Eighth Moment Conjecture96
(2.4)Remarks on Larsen's Eighth Moment Conjecture96
(2.5)How to apply Larsen's Eighth Moment Conjecture; its current status97
(2.6)Other tools to rule out finiteness of G[subscript geom]98
(2.7)Some conjectures on drops102
(2.8)More tools to rule out finiteness of G[subscript geom]: sheaves of perverse origin and their monodromy104
Chapter 3Additive character sums on A[superscript n]111
(3.1)The L[subscript psi] theorem111
(3.2)Proof of the L[subscript psi] Theorem 3.1.2112
(3.3)Interlude: the homothety contraction method113
(3.4)Return to the proof of the L[subscript psi] theorem122
(3.5)Monodromy of exponential sums of Deligne type on A[superscript n]123
(3.6)Interlude: an exponential sum calculation129
(3.7)Interlude: separation of variables136
(3.8)Return to the monodromy of exponential sums of Deligne type on A[superscript n]138
(3.9)Application to Deligne polynomials144
(3.10)Self dual families of Deligne polynomials146
(3.11)Proofs of the theorems on self dual families149
(3.12)Proof of Theorem 3.10.7156
(3.13)Proof of Theorem 3.10.9158
Chapter 4Additive character sums on more general X161
(4.1)The general setting161
(4.2)The perverse sheaf M(X, r, Z[subscript i]'s, e[subscript i]'s, [psi]) on P(e[subscript 1],...., e[subscript r])166
(4.3)Interlude An exponential sum identity174
(4.4)Return to the proof of Theorem 4.2.12178
(4.5)The subcases n=1 and n=2179
Chapter 5Multiplicative character sums on A[superscript n]185
(5.1)The general setting185
(5.2)First main theorem: the case when [chi superscript e] is nontrivial188
(5.3)Continuation of the proof of Theorem 5.2.2 for n=1191
(5.4)Continuation of the proof of Theorem 5.2.2 for general n200
(5.5)Analysis of Gr[superscript 0](m(n, e, [chi])), for e prime to p but [chi superscript e] = 1207
(5.6)Proof of Theorem 5.5.2 in the case n [greater than or equal] 2210
Chapter 6Middle additive convolution221
(6.1)Middle convolution and its effect on local monodromy221
(6.2)Interlude: some galois theory in one variable233
(6.3)Proof of Theorem 6.2.11240
(6.4)Interpretation in terms of Swan conductors245
(6.5)Middle convolution and purity248
(6.6)Application to the monodromy of multiplicative character sums in several variables253
(6.7)Proof of Theorem 6.6.5, and applications255
(6.8)Application to the monodromy of additive character sums in several variables270
Appendix A6Swan-minimal poles281
(A6.1)Swan conductors of direct images281
(A6.2)An application to Swan conductors of pullbacks285
(A6.3)Interpretation in terms of canonical extensions287
(A6.4)Belyi polynomials, non-canonical extensions, and hypergeometric sheaves291
Chapter 7Pullbacks to curves from A[superscript 1]295
(7.1)The general pullback setting295
(7.2)General results on G[subscript geom] for pullbacks303
(7.3)Application to pullback families of elliptic curves and of their symmetric powers308
(7.4)Cautionary examples312
(7.5)Appendix: Degeneration of Leray spectral sequences317
Chapter 8One variable twists on curves321
(8.1)Twist sheaves in the sense of [Ka-TLFM]321
(8.2)Monodromy of twist sheaves in the sense of [Ka-TLFM]324
Chapter 9Weierstrass sheaves as inputs327
(9.1)Weierstrass sheaves327
(9.2)The situation when 2 is invertible330
(9.3)Theorems of geometric irreducibility in odd characteristic331
(9.4)Geometric Irreducibility in even characteristic343
Chapter 10Weierstrass families349
(10.1)Universal Weierstrass families in arbitrary characteristic349
(10.2)Usual Weierstrass families in characteristic p [greater than or equal] 5356
Chapter 11FJTwist families and variants371
(11.1)(FJ, twist) families in characteristic p [greater than or equal] 5371
(11.2)(j[superscript -1], twist) families in characteristic 3380
(11.3)(j[superscript -1], twist) families in characteristic 2390
(11.4)End of the proof of 11.3.25: Proof that G[subscript geom] contains a reflection401
Chapter 12Uniformity results407
(12.1)Fibrewise perversity: basic properties407
(12.2)Uniformity results for monodromy; the basic setting409
(12.3)The Uniformity Theorem411
(12.4)Applications of the Uniformity Theorem to twist sheaves416
(12.5)Applications to multiplicative character sums421
(12.6)Non-application (sic!) to additive character sums427
(12.7)Application to generalized Weierstrass families of elliptic curves428
(12.8)Application to usual Weierstrass families of elliptic curves430
(12.9)Application to FJTwist families of elliptic curves433
(12.10)Applications to pullback families of elliptic curves435
(12.11)Application to quadratic twist families of elliptic curves439
Chapter 13Average analytic rank and large N limits443
(13.1)The basic setting443
(13.2)Passage to the large N limit: general results448
(13.3)Application to generalized Weierstrass families of elliptic curves449
(13.4)Application to usual Weierstrass families of elliptic curves450
(13.5)Applications to FJTwist families of elliptic curves451
(13.6)Applications to pullback families of elliptic curves452
(13.7)Applications to quadratic twist families of elliptic curves453
References455
Notation Index461
Subject Index467

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