Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas

Name: Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas | (AMS-212)
Brand: Princeton University Press
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(AMS-212)

Daniel Kriz(Autor*in)

Princeton University Press

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Erschienen am 9. November 2021

280 Seiten

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Inhalt

Cover
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
2. Preliminaries: Generalities
2.1 Grothendieck sites and topoi
2.2 Pro-categories
2.3 Adic spaces
2.4 (Pre-)adic spaces and (pre)perfectoid spaces
2.5 Some complements on inverse limits of (pre-)adic spaces
2.6 The proétale site of an adic space
2.7 Period sheaves
2.8 The proétale "constant sheaf" ?p,Y
2.9 B+ dR,Y -local systems, OB+ dR,Y -modules with connection, and the general de Rham comparison theorem
3. Preliminaries: Geometry of the infinite-level modular curve
3.1 The infinite-level modular curve
3.2 Relative étale cohomology and the Weil pairing
3.3 The GL2(Qp)-action on Y (and Y)
3.4 The Hodge-Tate period and the Hodge-Tate period map
3.5 The Lubin-Tate period on the supersingular locus
3.6 The relative Hodge-Tate filtration
3.7 The fake Hasse invariant
3.8 Relative de Rham cohomology and the Hodge-de Rham filtration
3.9 Relative p-adic de Rham comparison theorem applied to AY
4. The fundamental de Rham periods
4.1 A proétale local description of OB(+) dR
4.2 The fundamental de Rham periods
4.3 GL2(Qp)-transformation properties of the fundamental de Rham periods
4.4 The p-adic Legendre relation
4.5 Relation to Colmez's "p-adic period pairing"
4.6 Relation to classical (Serre-Tate) theory on the ordinary locus
4.7 The Kodaira-Spencer isomorphism
4.8 The fundamental de Rham period zdR
4.9 The canonical differential
5. Thep-adic Maass-Shimura operator
5.1 The "horizontal" lifting of the Hodge-Tate filtration
5.2 The "horizontal" relative Hodge-Tate decomposition over O?
5.3 Definition of the p-adic Maass-Shimura operator
5.4 The p-adic Maass-Shimura operator in coordinates and generalized p-adic modular forms
5.5 The p-adic Maass-Shimura operator with "nearly holomorphic coefficients"
5.6 The relative Hodge-Tate decomposition over O┼?
5.7 The p-adic Maass-Shimura operator in coordinates and generalized p-adic nearly holomorphic modular forms
5.8 Relation of djk and (d ┼ k)j to the ordinary Atkin-Serre operator djk,AS and Katz's p-adic modular forms
5.9 Comparison between the complex and p-adic Maass-Shimura operators at CM points
5.10 Comparison of algeraic Maass-Shimura derivatives on different levels
6. p-adic analysis of the p-adic Maass-Shimura operators
6.1 qdR-expansions
6.2 Relation between qdR-expansions and Serre-Tate expansions
6.3 Integrality properties of qdR-expansions: the Dieudonné-Dwork lemma
6.4 Integral structures on stalks of intermediate period sheaves between O? and Ô?
6.5 The p-adic Maass-Shimura operator Ojk in qdR-coordinates
6.6 Integrality of qdR-expansions and the b-operator
6.7 p-adic analytic properties of p-adic Maass-Shimura operators
7. Bounding periods at supersingular CM points
7.1 Periods of supersingular CM points
7.2 Weights
7.3 Good CM points
8. Supersingular Rankin-Selberg p-adic L-functions
8.1 Preliminaries for the construction
8.2 Construction of the p-adic L-function
8.3 Interpolation
8.3.1 Interpolation formula
8.3.2 Interpolation formula
9. The p-adic Waldspurger formula
9.1 Coleman integration
9.2 Coleman primitives in our situation
9.3 The p-adic Waldspurger formula
9.4 p-adic Kronecker limit formula
Bibliography
Index

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