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

## Overview

Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition into the twenty-first century as Princeton looks forward to publishing the major works of the new millennium. To mark the continued success of the series, all books are again available in paperback. For a complete list of titles, please visit the Princeton University Press Web site: pup.princeton.edu. The most recently published volumes include: Green's Function Estimates for Lattice Schrodinger Operators and Applications, by J. Bourgain, On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety, by Mark Green and Phillip Griffiths, Radon Transforms and the Rigidity of the Grassmannians, by Jacques Gasqui and Hubert Goldschmidt, Markov Processes from K. Ito's Perspective, by Daniel W. Stroock, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation, by Spyridon Kamvissis, Kenneth D. T-R. McLaughlin, and Peter D. Miller, Finite Structures with Few Types, by Gregory Cherlin and Ehud Hrushovski, The Geometry and Cohomology of Some Simple Shimura Varieties, by Michael Harris and Richard Taylor with an appendix by Vladimir G. Berkovich, Twisted L-Functions and Monodromy, by Nicholas M. Katz, Surveys on Surgery Theory: Volume 2. Papers Dedicated to C. T. C. Wall, edited by Sylvain Cappell, Andrew Ranicki, and Jonathan Rosenberg, Triangulated Categories, by Amnon Neeman.

## Product Details

ISBN-13: | 9780691123295 |
---|---|

Publisher: | Princeton University Press |

Publication date: | 09/12/2005 |

Series: | Annals of Mathematics Studies Series |

Pages: | 448 |

Product dimensions: | 7.12(w) x 10.24(h) x 1.47(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*.

## Read an Excerpt

**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*.

## First Chapter

**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 |

## Reading Group Guide

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 |

## Interviews

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 |

## Recipe

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) | 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) | 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 |