Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas:
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| Format: | Book |
| Language: | English |
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Princeton ; Oxford
Princeton University Press
[2021]
|
| Series: | Annals of Mathematics Studies
Number 212 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiii, 258 Seiten 24 cm |
| ISBN: | 9780691216461 0691216460 9780691216478 0691216479 |
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Record in the Search Index
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| adam_text | Supersingular p-adic L-functions,
Maass-Shimura Operators and
Waldspurger Formulas
Daniel J Kriz
PRINCETON UNIVERSITY PRESS -
PRINCETON AND OXFORD
Contents
Preface
Acknowledgments
1 Introduction
1 1 Previous constructions and Katz’s theory of p-adic
modular forms on the ordinary locus
1 2 Outline of our theory of p-adic analysis on the
supersingular locus and construction of p-adic
L-functions
1 3 Main results
1 4 Some remarks on other works in supersingular
Iwasawa theory
Preliminaries: Generalities
Grothendieck sites and topoi
Pro-categories
Adic spaces
(Pre-)adic spaces and (pre)perfectoid spaces
Some complements on inverse limits of (pre-)adic spaces
The proétale site of an adic space
Period sheaves
The proétale “constant sheaf” Zp,y
Bir y-local systems, OBiR,y-modules with connection,
and the general de Rham comparison theorem
Preliminaries: Geometry of the infinite-level
modular curve
The infinite-level modular curve
Relative étale cohomology and the Weil pairing
The GL2(Q,)-action on Y (and Y)
The Hodge-Tate period and the Hodge-Tate period map
The Lubin-Tate period on the supersingular locus
The relative Hodge-Tate filtration
The fake Hasse invariant
xi
xv
viii
CONTENTS
Relative de Rham cohomology and the Hodge-de
Rham filtration
Relative p-adic de Rham comparison theorem
applied to A+ Y
4 The fundamental de Rham periods
A proétale local description of oss)
The fundamental de Rham periods
GL2(Q,)-transformation properties of the fundamental
de Rham periods
The p-adic Legendre relation
Relation to Colmez’s “p-adic period pairing”
Relation to classical (Serre-Tate) theory on
the ordinary locus
The Kodaira-Spencer isomorphism
The fundamental de Rham period zur
The canonical differential
5 The p-adic Maass-Shimura operator
The “horizontal” lifting of the Hodge-Tate filtration
The “horizontal” relative Hodge-Tate decomposition
over On
Definition of the p-adic Maass-Shimura operator
The p-adic Maass-Shimura operator in coordinates and
generalized p-adic modular forms
The p-adic Maass-Shimura operator with “nearly
holomorphic coefficients”
The relative Hodge-Tate decomposition over ol
The p-adic Maass-Shimura operator in coordinates and
generalized p-adic nearly holomorphic modular forms
Relation of dj and (dj) to the ordinary Atkin-Serre
operator dj as and Katz’s p-adic modular forms
Comparison between the complex and p-adic
Maass-Shimura operators at CM points
Comparison of algeraic Maass-Shimura derivatives
on different levels
6 p-adic analysis of the p-adic Maass-Shimura operators
gar-eXpansions
Relation between gar-expansions and Serre-Tate expansions
Integrality properties of gar-expansions:
the Dieudonné-Dwork lemma
Integral structures on stalks of intermediate period
sheaves between O, and On
CONTENTS ix
6 5 The p-adic Maass-Shimura operator a in gar-coordinates 182
6 6 Integrality of gar-expansions and the b-operator 184
6 7 p-adic analytic properties of p-adic Maass-Shimura operators 186
7 Bounding periods at supersingular CM points 197
7 1 Periods of supersingular CM points 197
7 2 Weights 205
7 3 Good CM points 208
8 Supersingular Rankin-Selberg p-adic L-functions 216
8 1 Preliminaries for the construction 216
8 2 Construction of the p-adic L-function 219
8 3 Interpolation 228
831 Interpolation formula 228
832 Interpolation formula 234
9 The p-adic Waldspurger formula 236
9 1 Coleman integration 237
9 2 Coleman primitives in our situation 237
9 3 The p-adic Waldspurger formula 243
9 4 p-adic Kronecker limit formula 248
Bibliography 251
Index 257
|
| adam_txt |
Supersingular p-adic L-functions,
Maass-Shimura Operators and
Waldspurger Formulas
Daniel J Kriz
PRINCETON UNIVERSITY PRESS -
PRINCETON AND OXFORD
Contents
Preface
Acknowledgments
1 Introduction
1 1 Previous constructions and Katz’s theory of p-adic
modular forms on the ordinary locus
1 2 Outline of our theory of p-adic analysis on the
supersingular locus and construction of p-adic
L-functions
1 3 Main results
1 4 Some remarks on other works in supersingular
Iwasawa theory
Preliminaries: Generalities
Grothendieck sites and topoi
Pro-categories
Adic spaces
(Pre-)adic spaces and (pre)perfectoid spaces
Some complements on inverse limits of (pre-)adic spaces
The proétale site of an adic space
Period sheaves
The proétale “constant sheaf” Zp,y
Bir y-local systems, OBiR,y-modules with connection,
and the general de Rham comparison theorem
Preliminaries: Geometry of the infinite-level
modular curve
The infinite-level modular curve
Relative étale cohomology and the Weil pairing
The GL2(Q,)-action on Y (and Y)
The Hodge-Tate period and the Hodge-Tate period map
The Lubin-Tate period on the supersingular locus
The relative Hodge-Tate filtration
The fake Hasse invariant
xi
xv
viii
CONTENTS
Relative de Rham cohomology and the Hodge-de
Rham filtration
Relative p-adic de Rham comparison theorem
applied to A+ Y
4 The fundamental de Rham periods
A proétale local description of oss)
The fundamental de Rham periods
GL2(Q,)-transformation properties of the fundamental
de Rham periods
The p-adic Legendre relation
Relation to Colmez’s “p-adic period pairing”
Relation to classical (Serre-Tate) theory on
the ordinary locus
The Kodaira-Spencer isomorphism
The fundamental de Rham period zur
The canonical differential
5 The p-adic Maass-Shimura operator
The “horizontal” lifting of the Hodge-Tate filtration
The “horizontal” relative Hodge-Tate decomposition
over On
Definition of the p-adic Maass-Shimura operator
The p-adic Maass-Shimura operator in coordinates and
generalized p-adic modular forms
The p-adic Maass-Shimura operator with “nearly
holomorphic coefficients”
The relative Hodge-Tate decomposition over ol
The p-adic Maass-Shimura operator in coordinates and
generalized p-adic nearly holomorphic modular forms
Relation of dj and (dj) to the ordinary Atkin-Serre
operator dj as and Katz’s p-adic modular forms
Comparison between the complex and p-adic
Maass-Shimura operators at CM points
Comparison of algeraic Maass-Shimura derivatives
on different levels
6 p-adic analysis of the p-adic Maass-Shimura operators
gar-eXpansions
Relation between gar-expansions and Serre-Tate expansions
Integrality properties of gar-expansions:
the Dieudonné-Dwork lemma
Integral structures on stalks of intermediate period
sheaves between O, and On
CONTENTS ix
6 5 The p-adic Maass-Shimura operator a in gar-coordinates 182
6 6 Integrality of gar-expansions and the b-operator 184
6 7 p-adic analytic properties of p-adic Maass-Shimura operators 186
7 Bounding periods at supersingular CM points 197
7 1 Periods of supersingular CM points 197
7 2 Weights 205
7 3 Good CM points 208
8 Supersingular Rankin-Selberg p-adic L-functions 216
8 1 Preliminaries for the construction 216
8 2 Construction of the p-adic L-function 219
8 3 Interpolation 228
831 Interpolation formula 228
832 Interpolation formula 234
9 The p-adic Waldspurger formula 236
9 1 Coleman integration 237
9 2 Coleman primitives in our situation 237
9 3 The p-adic Waldspurger formula 243
9 4 p-adic Kronecker limit formula 248
Bibliography 251
Index 257 |
| any_adam_object | 1 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-03T19:02:51Z |
| indexdate | 2024-07-10T09:21:46Z |
| institution | BVB |
| isbn | 9780691216461 0691216460 9780691216478 0691216479 |
| language | English |
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| publisher | Princeton University Press |
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| series | Annals of Mathematics Studies |
| series2 | Annals of Mathematics Studies |
| spelling | Kriz, Daniel J. 1992- Verfasser (DE-588)1247640930 aut Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas Daniel J. Kriz Princeton ; Oxford Princeton University Press [2021] xiii, 258 Seiten 24 cm txt rdacontent n rdamedia nc rdacarrier Annals of Mathematics Studies Number 212 Number theory p-adic analysis L-functions 9780691225739 (e-book) 0691225737 (e-book) Erscheint auch als Online-Ausgabe Kriz, Daniel Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas Princeton : Princeton University Press, 2021 0691225737 Annals of Mathematics Studies Number 212 (DE-604)BV000000991 212 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033186435&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kriz, Daniel J. 1992- Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas Annals of Mathematics Studies |
| title | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas |
| title_auth | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas |
| title_exact_search | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas |
| title_exact_search_txtP | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas |
| title_full | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas Daniel J. Kriz |
| title_fullStr | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas Daniel J. Kriz |
| title_full_unstemmed | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas Daniel J. Kriz |
| title_short | Supersingular p-adic L-functions, Maass-Shimura operators and Waldspurger formulas |
| title_sort | supersingular p adic l functions maass shimura operators and waldspurger formulas |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033186435&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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