skip to content
Moments, monodromy, and perversity : a diophantine perspective Preview this item
ClosePreview this item
Checking...

Moments, monodromy, and perversity : a diophantine perspective

Author: Nicholas M Katz
Publisher: Princeton : Princeton University Press, 2005.
Series: Annals of mathematics studies, no. 159.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Develops techniques based on two ingredients: the theory of perverse sheaves and Larsen's Alternative. These techniques are then used to calculate the geometric monodromy groups attached to some  Read more...

Rating:

(not yet rated) 0 with reviews - Be the first.

Subjects
More like this

 

Find a copy online

Links to this item

Find a copy in the library

&AllPage.SpinnerRetrieving; Finding libraries that hold this item...

Details

Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Nicholas M Katz
ISBN: 0691123292 0691123306 9780691123295 9780691123301
OCLC Number: 56924745
Description: viii, 475 pages ; 26 cm.
Contents: 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 [characters not reproducible] 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.
Series Title: Annals of mathematics studies, no. 159.
Responsibility: Nicholas M. Katz.
More information:

Reviews

User-contributed reviews
Retrieving GoodReads reviews...
Retrieving DOGObooks reviews...

Tags

Be the first.
Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

Linked Data


Primary Entity

<http://www.worldcat.org/oclc/56924745> # Moments, monodromy, and perversity : a diophantine perspective
    a schema:CreativeWork, schema:Book ;
    library:oclcnum "56924745" ;
    library:placeOfPublication <http://id.loc.gov/vocabulary/countries/nju> ;
    library:placeOfPublication <http://experiment.worldcat.org/entity/work/data/326516364#Place/princeton> ; # Princeton
    schema:about <http://experiment.worldcat.org/entity/work/data/326516364#Topic/faisceaux_theorie_des> ; # Faisceaux, Théorie des
    schema:about <http://experiment.worldcat.org/entity/work/data/326516364#Topic/monodromiegruppe> ; # Monodromiegruppe
    schema:about <http://experiment.worldcat.org/entity/work/data/326516364#Topic/garbentheorie> ; # Garbentheorie
    schema:about <http://id.worldcat.org/fast/1025575> ; # Monodromy groups
    schema:about <http://id.worldcat.org/fast/989693> ; # L-functions
    schema:about <http://id.worldcat.org/fast/1115421> ; # Sheaf theory
    schema:about <http://experiment.worldcat.org/entity/work/data/326516364#Topic/fonctions_l> ; # Fonctions L
    schema:about <http://experiment.worldcat.org/entity/work/data/326516364#Topic/groupes_de_monodromie> ; # Groupes de monodromie
    schema:about <http://dewey.info/class/512.73/e22/> ;
    schema:about <http://experiment.worldcat.org/entity/work/data/326516364#Topic/l_funktion> ; # L-Funktion
    schema:bookFormat bgn:PrintBook ;
    schema:creator <http://viaf.org/viaf/24649612> ; # Nicholas M. Katz
    schema:datePublished "2005" ;
    schema:exampleOfWork <http://worldcat.org/entity/work/id/326516364> ;
    schema:inLanguage "en" ;
    schema:isPartOf <http://experiment.worldcat.org/entity/work/data/326516364#Series/annals_of_mathematics_studies> ; # Annals of mathematics studies ;
    schema:name "Moments, monodromy, and perversity : a diophantine perspective"@en ;
    schema:productID "56924745" ;
    schema:publication <http://www.worldcat.org/title/-/oclc/56924745#PublicationEvent/princeton_princeton_university_press_2005> ;
    schema:publisher <http://experiment.worldcat.org/entity/work/data/326516364#Agent/princeton_university_press> ; # Princeton University Press
    schema:url <http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014186569&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA> ;
    schema:url <http://www.gbv.de/dms/goettingen/470937661.pdf> ;
    schema:url <http://catdir.loc.gov/catdir/enhancements/fy0654/2004062828-t.html> ;
    schema:url <http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=014186569&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA> ;
    schema:workExample <http://worldcat.org/isbn/9780691123295> ;
    schema:workExample <http://worldcat.org/isbn/9780691123301> ;
    umbel:isLike <http://bnb.data.bl.uk/id/resource/GBA573553> ;
    wdrs:describedby <http://www.worldcat.org/title/-/oclc/56924745> ;
    .


Related Entities

<http://experiment.worldcat.org/entity/work/data/326516364#Agent/princeton_university_press> # Princeton University Press
    a bgn:Agent ;
    schema:name "Princeton University Press" ;
    .

<http://experiment.worldcat.org/entity/work/data/326516364#Series/annals_of_mathematics_studies> # Annals of mathematics studies ;
    a bgn:PublicationSeries ;
    schema:hasPart <http://www.worldcat.org/oclc/56924745> ; # Moments, monodromy, and perversity : a diophantine perspective
    schema:name "Annals of mathematics studies ;" ;
    .

<http://experiment.worldcat.org/entity/work/data/326516364#Topic/faisceaux_theorie_des> # Faisceaux, Théorie des
    a schema:Intangible ;
    schema:name "Faisceaux, Théorie des"@fr ;
    .

<http://experiment.worldcat.org/entity/work/data/326516364#Topic/groupes_de_monodromie> # Groupes de monodromie
    a schema:Intangible ;
    schema:name "Groupes de monodromie"@fr ;
    .

<http://id.worldcat.org/fast/1025575> # Monodromy groups
    a schema:Intangible ;
    schema:name "Monodromy groups"@en ;
    .

<http://id.worldcat.org/fast/1115421> # Sheaf theory
    a schema:Intangible ;
    schema:name "Sheaf theory"@en ;
    .

<http://id.worldcat.org/fast/989693> # L-functions
    a schema:Intangible ;
    schema:name "L-functions"@en ;
    .

<http://viaf.org/viaf/24649612> # Nicholas M. Katz
    a schema:Person ;
    schema:birthDate "1943" ;
    schema:familyName "Katz" ;
    schema:givenName "Nicholas M." ;
    schema:name "Nicholas M. Katz" ;
    .

<http://worldcat.org/isbn/9780691123295>
    a schema:ProductModel ;
    schema:isbn "0691123292" ;
    schema:isbn "9780691123295" ;
    .

<http://worldcat.org/isbn/9780691123301>
    a schema:ProductModel ;
    schema:isbn "0691123306" ;
    schema:isbn "9780691123301" ;
    .


Content-negotiable representations

Close Window

Please sign in to WorldCat 

Don't have an account? You can easily create a free account.