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

Moments, Monodromy, and Perversity. (AM-159)

A Diophantine Perspective. (AM-159)

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Published: 12th September 2005
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It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions). Roughly speaking, Deligne showed that any such family obeys a "generalized Sato-Tate law," and that figuring out which generalized Sato-Tate law applies to a given family amounts essentially to computing a certain complex semisimple (not necessarily connected) algebraic group, the "geometric monodromy group" attached to that family.

Up to now, nearly all techniques for determining geometric monodromy groups have relied, at least in part, on local information. In "Moments, Monodromy, and Perversity," Nicholas Katz develops new techniques, which are resolutely global in nature. They are based on two vital ingredients, neither of which existed at the time of Deligne's original work on the subject. The first is the theory of perverse sheaves, pioneered by Goresky and MacPherson in the topological setting and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber. The second is Larsen's Alternative, which very nearly characterizes classical groups by their fourth moments. These new techniques, which are of great interest in their own right, are first developed and then used to calculate the geometric monodromy groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields.

Introductionp. 1
Basic results on perversity and higher momentsp. 9
The notion of a d-separating space of functionsp. 9
Review of semiperversity and perversityp. 12
A twisting construction: the object Twist(L,K,F,h)p. 13
The basic theorem and its consequencesp. 13
Review of weightsp. 21
Remarks on the various notions of mixednessp. 24
The Orthogonality Theoremp. 25
First Applications of the Orthogonality Theoremp. 31
Questions of autoduality: the Frobenius-Schur indicator theoremp. 36
Dividing out the "constant part" of an [iota]-pure perverse sheafp. 42
The subsheaf N[subscript ncst0] in the mixed casep. 44
Interlude: abstract trace functions; approximate trace functionsp. 45
Two uniqueness theoremsp. 47
The central normalization F[subscript 0] of a trace function Fp. 50
First applications to the objects Twist(L, K, F, h): The notion of standard inputp. 52
Review of higher momentsp. 60
Higher moments for geometrically irreducible lisse sheavesp. 61
Higher moments for geometrically irreducible perverse sheavesp. 62
A fundamental inequalityp. 62
Higher moment estimates for Twist(L,K,F,h)p. 64
Proof of the Higher Moment Theorem 1.20.2: combinatorial preliminariesp. 67
Variations on the Higher Moment Theoremp. 76
Counterexamplesp. 87
How to apply the results of Chapter 1p. 93
How to apply the Higher Moment Theoremp. 93
Larsen's Alternativep. 94
Larsen's Eighth Moment Conjecturep. 96
Remarks on Larsen's Eighth Moment Conjecturep. 96
How to apply Larsen's Eighth Moment Conjecture; its current statusp. 97
Other tools to rule out finiteness of G[subscript geom]p. 98
Some conjectures on dropsp. 102
More tools to rule out finiteness of G[subscript geom]: sheaves of perverse origin and their monodromyp. 104
Additive character sums on A[superscript n]p. 111
The L[subscript psi] theoremp. 111
Proof of the L[subscript psi] Theorem 3.1.2p. 112
Interlude: the homothety contraction methodp. 113
Return to the proof of the L[subscript psi] theoremp. 122
Monodromy of exponential sums of Deligne type on A[superscript n]p. 123
Interlude: an exponential sum calculationp. 129
Interlude: separation of variablesp. 136
Return to the monodromy of exponential sums of Deligne type on A[superscript n]p. 138
Application to Deligne polynomialsp. 144
Self dual families of Deligne polynomialsp. 146
Proofs of the theorems on self dual familiesp. 149
Proof of Theorem 3.10.7p. 156
Proof of Theorem 3.10.9p. 158
Additive character sums on more general Xp. 161
The general settingp. 161
The perverse sheaf M(X, r, Z[subscript i]'s, e[subscript i]'s, [psi]) on P(e[subscript 1],...., e[subscript r])p. 166
Interlude An exponential sum identityp. 174
Return to the proof of Theorem 4.2.12p. 178
The subcases n=1 and n=2p. 179
Multiplicative character sums on A[superscript n]p. 185
The general settingp. 185
First main theorem: the case when [chi superscript e] is nontrivialp. 188
Continuation of the proof of Theorem 5.2.2 for n=1p. 191
Continuation of the proof of Theorem 5.2.2 for general np. 200
Analysis of Gr[superscript 0](m(n, e, [chi])), for e prime to p but [chi superscript e] = 1p. 207
Proof of Theorem 5.5.2 in the case n [greater than or equal] 2p. 210
Middle additive convolutionp. 221
Middle convolution and its effect on local monodromyp. 221
Interlude: some galois theory in one variablep. 233
Proof of Theorem 6.2.11p. 240
Interpretation in terms of Swan conductorsp. 245
Middle convolution and purityp. 248
Application to the monodromy of multiplicative character sums in several variablesp. 253
Proof of Theorem 6.6.5, and applicationsp. 255
Application to the monodromy of additive character sums in several variablesp. 270
Swan-minimal polesp. 281
Swan conductors of direct imagesp. 281
An application to Swan conductors of pullbacksp. 285
Interpretation in terms of canonical extensionsp. 287
Belyi polynomials, non-canonical extensions, and hypergeometric sheavesp. 291
Pullbacks to curves from A[superscript 1]p. 295
The general pullback settingp. 295
General results on G[subscript geom] for pullbacksp. 303
Application to pullback families of elliptic curves and of their symmetric powersp. 308
Cautionary examplesp. 312
Appendix: Degeneration of Leray spectral sequencesp. 317
One variable twists on curvesp. 321
Twist sheaves in the sense of [Ka-TLFM]p. 321
Monodromy of twist sheaves in the sense of [Ka-TLFM]p. 324
Weierstrass sheaves as inputsp. 327
Weierstrass sheavesp. 327
The situation when 2 is invertiblep. 330
Theorems of geometric irreducibility in odd characteristicp. 331
Geometric Irreducibility in even characteristicp. 343
Weierstrass familiesp. 349
Universal Weierstrass families in arbitrary characteristicp. 349
Usual Weierstrass families in characteristic p [greater than or equal] 5p. 356
FJTwist families and variantsp. 371
(FJ, twist) families in characteristic p [greater than or equal] 5p. 371
(j[superscript -1], twist) families in characteristic 3p. 380
(j[superscript -1], twist) families in characteristic 2p. 390
End of the proof of 11.3.25: Proof that G[subscript geom] contains a reflectionp. 401
Uniformity resultsp. 407
Fibrewise perversity: basic propertiesp. 407
Uniformity results for monodromy; the basic settingp. 409
The Uniformity Theoremp. 411
Applications of the Uniformity Theorem to twist sheavesp. 416
Applications to multiplicative character sumsp. 421
Non-application (sic!) to additive character sumsp. 427
Application to generalized Weierstrass families of elliptic curvesp. 428
Application to usual Weierstrass families of elliptic curvesp. 430
Application to FJTwist families of elliptic curvesp. 433
Applications to pullback families of elliptic curvesp. 435
Application to quadratic twist families of elliptic curvesp. 439
Average analytic rank and large N limitsp. 443
The basic settingp. 443
Passage to the large N limit: general resultsp. 448
Application to generalized Weierstrass families of elliptic curvesp. 449
Application to usual Weierstrass families of elliptic curvesp. 450
Applications to FJTwist families of elliptic curvesp. 451
Applications to pullback families of elliptic curvesp. 452
Applications to quadratic twist families of elliptic curvesp. 453
Referencesp. 455
Notation Indexp. 461
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691123301
ISBN-10: 0691123306
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 475
Published: 12th September 2005
Country of Publication: US
Dimensions (cm): 24.99 x 17.68  x 1.63
Weight (kg): 0.4