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

by Nicholas M. Katz

Overview

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.

Product Details

ISBN-13:	9780691123301
Publisher:	Princeton University Press
Publication date:	10/02/2005
Series:	Annals of Mathematics Studies , #159
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

	Introduction	1
Chapter 1	Basic results on perversity and higher moments	9
(1.1)	The notion of a d-separating space of functions	9
(1.2)	Review of semiperversity and perversity	12
(1.3)	A twisting construction: the object Twist(L,K,F,h)	13
(1.4)	The basic theorem and its consequences	13
(1.5)	Review of weights	21
(1.6)	Remarks on the various notions of mixedness	24
(1.7)	The Orthogonality Theorem	25
(1.8)	First Applications of the Orthogonality Theorem	31
(1.9)	Questions of autoduality: the Frobenius-Schur indicator theorem	36
(1.10)	Dividing out the "constant part" of an [iota]-pure perverse sheaf	42
(1.11)	The subsheaf N[subscript ncst0] in the mixed case	44
(1.12)	Interlude: abstract trace functions; approximate trace functions	45
(1.13)	Two uniqueness theorems	47
(1.14)	The central normalization F[subscript 0] of a trace function F	50
(1.15)	First applications to the objects Twist(L, K, F, h): The notion of standard input	52
(1.16)	Review of higher moments	60
(1.17)	Higher moments for geometrically irreducible lisse sheaves	61
(1.18)	Higher moments for geometrically irreducible perverse sheaves	62
(1.19)	A fundamental inequality	62
(1.20)	Higher moment estimates for Twist(L,K,F,h)	64
(1.21)	Proof of the Higher Moment Theorem 1.20.2: combinatorial preliminaries	67
(1.22)	Variations on the Higher Moment Theorem	76
(1.23)	Counterexamples	87
Chapter 2	How to apply the results of Chapter 1	93
(2.1)	How to apply the Higher Moment Theorem	93
(2.2)	Larsen's Alternative	94
(2.3)	Larsen's Eighth Moment Conjecture	96
(2.4)	Remarks on Larsen's Eighth Moment Conjecture	96
(2.5)	How to apply Larsen's Eighth Moment Conjecture; its current status	97
(2.6)	Other tools to rule out finiteness of G[subscript geom]	98
(2.7)	Some conjectures on drops	102
(2.8)	More tools to rule out finiteness of G[subscript geom]: sheaves of perverse origin and their monodromy	104
Chapter 3	Additive character sums on A[superscript n]	111
(3.1)	The L[subscript psi] theorem	111
(3.2)	Proof of the L[subscript psi] Theorem 3.1.2	112
(3.3)	Interlude: the homothety contraction method	113
(3.4)	Return to the proof of the L[subscript psi] theorem	122
(3.5)	Monodromy of exponential sums of Deligne type on A[superscript n]	123
(3.6)	Interlude: an exponential sum calculation	129
(3.7)	Interlude: separation of variables	136
(3.8)	Return to the monodromy of exponential sums of Deligne type on A[superscript n]	138
(3.9)	Application to Deligne polynomials	144
(3.10)	Self dual families of Deligne polynomials	146
(3.11)	Proofs of the theorems on self dual families	149
(3.12)	Proof of Theorem 3.10.7	156
(3.13)	Proof of Theorem 3.10.9	158
Chapter 4	Additive character sums on more general X	161
(4.1)	The general setting	161
(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 identity	174
(4.4)	Return to the proof of Theorem 4.2.12	178
(4.5)	The subcases n=1 and n=2	179
Chapter 5	Multiplicative character sums on A[superscript n]	185
(5.1)	The general setting	185
(5.2)	First main theorem: the case when [chi superscript e] is nontrivial	188
(5.3)	Continuation of the proof of Theorem 5.2.2 for n=1	191
(5.4)	Continuation of the proof of Theorem 5.2.2 for general n	200
(5.5)	Analysis of Gr[superscript 0](m(n, e, [chi])), for e prime to p but [chi superscript e] = 1	207
(5.6)	Proof of Theorem 5.5.2 in the case n [greater than or equal] 2	210
Chapter 6	Middle additive convolution	221
(6.1)	Middle convolution and its effect on local monodromy	221
(6.2)	Interlude: some galois theory in one variable	233
(6.3)	Proof of Theorem 6.2.11	240
(6.4)	Interpretation in terms of Swan conductors	245
(6.5)	Middle convolution and purity	248
(6.6)	Application to the monodromy of multiplicative character sums in several variables	253
(6.7)	Proof of Theorem 6.6.5, and applications	255
(6.8)	Application to the monodromy of additive character sums in several variables	270
Appendix A6	Swan-minimal poles	281
(A6.1)	Swan conductors of direct images	281
(A6.2)	An application to Swan conductors of pullbacks	285
(A6.3)	Interpretation in terms of canonical extensions	287
(A6.4)	Belyi polynomials, non-canonical extensions, and hypergeometric sheaves	291
Chapter 7	Pullbacks to curves from A[superscript 1]	295
(7.1)	The general pullback setting	295
(7.2)	General results on G[subscript geom] for pullbacks	303
(7.3)	Application to pullback families of elliptic curves and of their symmetric powers	308
(7.4)	Cautionary examples	312
(7.5)	Appendix: Degeneration of Leray spectral sequences	317
Chapter 8	One variable twists on curves	321
(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 9	Weierstrass sheaves as inputs	327
(9.1)	Weierstrass sheaves	327
(9.2)	The situation when 2 is invertible	330
(9.3)	Theorems of geometric irreducibility in odd characteristic	331
(9.4)	Geometric Irreducibility in even characteristic	343
Chapter 10	Weierstrass families	349
(10.1)	Universal Weierstrass families in arbitrary characteristic	349
(10.2)	Usual Weierstrass families in characteristic p [greater than or equal] 5	356
Chapter 11	FJTwist families and variants	371
(11.1)	(FJ, twist) families in characteristic p [greater than or equal] 5	371
(11.2)	(j[superscript -1], twist) families in characteristic 3	380
(11.3)	(j[superscript -1], twist) families in characteristic 2	390
(11.4)	End of the proof of 11.3.25: Proof that G[subscript geom] contains a reflection	401
Chapter 12	Uniformity results	407
(12.1)	Fibrewise perversity: basic properties	407
(12.2)	Uniformity results for monodromy; the basic setting	409
(12.3)	The Uniformity Theorem	411
(12.4)	Applications of the Uniformity Theorem to twist sheaves	416
(12.5)	Applications to multiplicative character sums	421
(12.6)	Non-application (sic!) to additive character sums	427
(12.7)	Application to generalized Weierstrass families of elliptic curves	428
(12.8)	Application to usual Weierstrass families of elliptic curves	430
(12.9)	Application to FJTwist families of elliptic curves	433
(12.10)	Applications to pullback families of elliptic curves	435
(12.11)	Application to quadratic twist families of elliptic curves	439
Chapter 13	Average analytic rank and large N limits	443
(13.1)	The basic setting	443
(13.2)	Passage to the large N limit: general results	448
(13.3)	Application to generalized Weierstrass families of elliptic curves	449
(13.4)	Application to usual Weierstrass families of elliptic curves	450
(13.5)	Applications to FJTwist families of elliptic curves	451
(13.6)	Applications to pullback families of elliptic curves	452
(13.7)	Applications to quadratic twist families of elliptic curves	453
	References	455
	Notation Index	461
	Subject Index	467