Moments, monodromy, and perversity: a diophantine perspective
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, NJ [u.a.]
Princeton Univ. Press
2005
|
| Series: | Annals of mathematics studies
159 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 475 S. |
| ISBN: | 0691123292 0691123306 |
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| 245 | 1 | 0 | |a Moments, monodromy, and perversity |b a diophantine perspective |c Nicholas M. Katz |
| 264 | 1 | |a Princeton, NJ [u.a.] |b Princeton Univ. Press |c 2005 | |
| 300 | |a VIII, 475 S. | ||
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| adam_text | 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, 3r,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 i pure
perverse sheaf 42
(1.11) The subsheaf Nncsto 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 Fq 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,T,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
vi Contents
(2.5) How to apply Larsen s Eighth Moment Conjecture;
its current status 97
(2.6) Other tools to rule out finiteness of Ggeom 98
(2.7) Some conjectures on drops 102
(2.8) More tools to rule out finiteness of Ggeom: sheaves of
perverse origin and their monodromy 104
Chapter 3: Additive character sums on An 111
(3.1) The Lq, theorem 111
(3.2) Proof of the C^ Theorem 3.1.2 112
(3.3) Interlude: the homothety contraction method 113
(3.4) Return to the proof of the £jj theorem 122
(3.5) Monodromy of exponential sums of Deligne type on An 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 An 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; s, ej s, ijj) on lP(ei e ) 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 = l and n = 2 179
Chapter 5: Multiplicative character sums on An 185
(5.1) The general setting 185
(5.2) First main theorem: the case when %e is nontrivial 188
(5.3) Continuation of the proof of Theorem 5.2.2 for n = l 191
(5.4) Continuation of the proof of Theorem 5.2.2 for general n 200
(5.5) Analysis of Gr°(m(n, e, %)), for e prime to p but %e = 11 207
(5.6) Proof of Theorem 5.5.2 in the case n 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
Contents vii
(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^ 295
(7.1) The general pullback setting 295
(7.2) General results on Ggeom 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 5 356
Chapter 11: FJTwist families and variants 371
(11.1) (FJ, twist) families in characteristic p 5 371
(11.2) (j 1, twist) families in characteristic 3 380
(11.3) (j 1, twist) families in characteristic 2 390
viii Contents
(11.4) End of the proof of 11.3.25: Proof that Ggeom
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 Weierbtrass 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
|
| adam_txt |
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, 3r,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 i pure
perverse sheaf 42
(1.11) The subsheaf Nncsto 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 Fq 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,T,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
vi Contents
(2.5) How to apply Larsen's Eighth Moment Conjecture;
its current status 97
(2.6) Other tools to rule out finiteness of Ggeom 98
(2.7) Some conjectures on drops 102
(2.8) More tools to rule out finiteness of Ggeom: sheaves of
perverse origin and their monodromy 104
Chapter 3: Additive character sums on An 111
(3.1) The Lq, theorem 111
(3.2) Proof of the C^ Theorem 3.1.2 112
(3.3) Interlude: the homothety contraction method 113
(3.4) Return to the proof of the £jj theorem 122
(3.5) Monodromy of exponential sums of Deligne type on An 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 An 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;'s, ej's, ijj) on lP(ei e ) 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 = l and n = 2 179
Chapter 5: Multiplicative character sums on An 185
(5.1) The general setting 185
(5.2) First main theorem: the case when %e is nontrivial 188
(5.3) Continuation of the proof of Theorem 5.2.2 for n = l 191
(5.4) Continuation of the proof of Theorem 5.2.2 for general n 200
(5.5) Analysis of Gr°(m(n, e, %)), for e prime to p but %e = 11 207
(5.6) Proof of Theorem 5.5.2 in the case n 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
Contents vii
(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^ 295
(7.1) The general pullback setting 295
(7.2) General results on Ggeom 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 5 356
Chapter 11: FJTwist families and variants 371
(11.1) (FJ, twist) families in characteristic p 5 371
(11.2) (j"1, twist) families in characteristic 3 380
(11.3) (j"1, twist) families in characteristic 2 390
viii Contents
(11.4) End of the proof of 11.3.25: Proof that Ggeom
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 Weierbtrass 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 |
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| index_date | 2024-07-02T13:24:06Z |
| indexdate | 2024-07-09T20:26:58Z |
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| series | Annals of mathematics studies |
| series2 | Annals of mathematics studies |
| spelling | Katz, Nicholas M. 1943- Verfasser (DE-588)141265558 aut Moments, monodromy, and perversity a diophantine perspective Nicholas M. Katz Princeton, NJ [u.a.] Princeton Univ. Press 2005 VIII, 475 S. txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies 159 Faisceaux, Théorie des Fonctions L Groupes de monodromie L-functions Monodromy groups Sheaf theory Monodromiegruppe (DE-588)4194644-3 gnd rswk-swf Garbentheorie (DE-588)4155956-3 gnd rswk-swf L-Funktion (DE-588)4137026-0 gnd rswk-swf Monodromiegruppe (DE-588)4194644-3 s Garbentheorie (DE-588)4155956-3 s L-Funktion (DE-588)4137026-0 s DE-604 Annals of mathematics studies 159 (DE-604)BV000000991 159 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014186569&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Katz, Nicholas M. 1943- Moments, monodromy, and perversity a diophantine perspective Annals of mathematics studies Faisceaux, Théorie des Fonctions L Groupes de monodromie L-functions Monodromy groups Sheaf theory Monodromiegruppe (DE-588)4194644-3 gnd Garbentheorie (DE-588)4155956-3 gnd L-Funktion (DE-588)4137026-0 gnd |
| subject_GND | (DE-588)4194644-3 (DE-588)4155956-3 (DE-588)4137026-0 |
| title | Moments, monodromy, and perversity a diophantine perspective |
| title_auth | Moments, monodromy, and perversity a diophantine perspective |
| title_exact_search | Moments, monodromy, and perversity a diophantine perspective |
| title_exact_search_txtP | Moments, monodromy, and perversity a diophantine perspective |
| title_full | Moments, monodromy, and perversity a diophantine perspective Nicholas M. Katz |
| title_fullStr | Moments, monodromy, and perversity a diophantine perspective Nicholas M. Katz |
| title_full_unstemmed | Moments, monodromy, and perversity a diophantine perspective Nicholas M. Katz |
| title_short | Moments, monodromy, and perversity |
| title_sort | moments monodromy and perversity a diophantine perspective |
| title_sub | a diophantine perspective |
| topic | Faisceaux, Théorie des Fonctions L Groupes de monodromie L-functions Monodromy groups Sheaf theory Monodromiegruppe (DE-588)4194644-3 gnd Garbentheorie (DE-588)4155956-3 gnd L-Funktion (DE-588)4137026-0 gnd |
| topic_facet | Faisceaux, Théorie des Fonctions L Groupes de monodromie L-functions Monodromy groups Sheaf theory Monodromiegruppe Garbentheorie L-Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014186569&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000991 |
| work_keys_str_mv | AT katznicholasm momentsmonodromyandperversityadiophantineperspective |