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p-Adic Automorphic Forms on Shimura Varieties 2004 ed. [Kietas viršelis]

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  • Formatas: Hardback, 390 pages, aukštis x plotis: 235x155 mm, weight: 804 g, XI, 390 p., 1 Hardback
  • Serija: Springer Monographs in Mathematics
  • Išleidimo metai: 10-May-2004
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 0387207112
  • ISBN-13: 9780387207117
Kitos knygos pagal šią temą:
p-Adic Automorphic Forms on Shimura Varieties 2004 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 390 pages, aukštis x plotis: 235x155 mm, weight: 804 g, XI, 390 p., 1 Hardback
  • Serija: Springer Monographs in Mathematics
  • Išleidimo metai: 10-May-2004
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 0387207112
  • ISBN-13: 9780387207117
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780387207117.html
Raktažodžiai:
The book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research. The area covered in this book is now a focal point of research worldwide with many far-reaching applications that have led to solutions of longstanding problems and conjectures. Specifically, the use of p-adic elliptic and Hilbert modular forms have proven essential in recent breakthroughs in number theory.

This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry:1. An elementary construction of Shimura varieties as moduli of abelian schemes2. p-adic deformation theory of automorphic forms on Shimura varieties3. A simple proof of irreducibility of the generalized Igusa tower over the Shimura varietyThe book starts with a detailed study of elliptic and Hilbert modular forms and reaches to the forefront of research of Shimura varieties associated with general classical groups. The method of constructing p-adic analytic families and the proof of irreducibility was recently discovered by the author. The area covered in this book is now a focal point of research worldwide with many far-reaching applications that have led to solutions of longstanding problems and conjectures. Specifically, the use of p-adic elliptic and Hilbert modular forms have proven essential in recent breakthroughs in number theory (for example, the proof of Fermat's Last Theorem and the Shimura-Taniyama conjecture by A. Wiles and others).Haruzo Hida is Professor of Mathematics at University of California, Los Angeles. His previous books include Modular Forms and Galois Cohomology (Cambridge University Press 2000) and Geometric Modular Forms and Elliptic Curves (World Scientific Publishing Company 2000).

Recenzijos

From the reviews:









"Hida views the study of the geometric Galois group of the Shimura tower, as a geometric reciprocity law . general goal of the book is to incorporate Shimuras reciprocity law in a broader scheme of integral reciprocity laws which includes Iwasawa theory in its scope. a beautiful and very useful reference for anybody interested in the arithmetic theory of automorphic forms." (Jacques Tilouine, Mathematical Reviews, 2005e)



"The first purpose of this book is to supply the base of the construction of the Shimura variety. The second one is to introduce integrality of automorphic forms on such varieties . The mathematics discussed here is wonderful but highly nontrivial. The book will certainly be useful to graduate students and researchers entering this beautiful and difficult area of research." (Andrzej Dabrowski, Zentralblatt MATH, Vol. 1055, 2005)



"The purpose of this book is twofold: First to establish a p-adic deformationtheory of automorphic forms on Shimura varieties; this is recent work of the author. Second, to explain some of the necessary background, in particular the theory of moduli and Shimura varieties of PEL type . The book requires some familiarity with algebraic number theory and algebraic geometry (schemes) but is rather complete in the details. Thus, it may also serve as an introduction to Shimura varieties as well as their deformation theory." (J. Mahnkopf, Monatshefte für Mathematik, Vol. 146 (4), 2005)



"The idea is to study the p-adic variation of automorphic forms. This book is a high-level exposition of the theory for automorphic forms on Shimura Varieties. It includes a discussion of the special cases of elliptic modular forms and Hilbert modular forms, so it will be a useful resource for those wanting to learn the subject. The exposition is very dense, however, and the prerequisites are extensive. Overall, this is a book I am happy to have on my shelves ." (Fernando Q. Gouvźa, Math DL, January, 2004)



"Hida showed that ordinary p-adic modular forms moved naturally in p-adic families. In the book under review Hida has returned to the geometric construction of p-adic families of ordinary forms. Hidas theory has had many applications in the theory of classical modular forms, and as mathematics continues to mature, this more general theory will no doubt have similarly striking applications in the theory of automorphic forms." (K. Buzzard, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 109 (4), 2007)

1 Introduction 1(16)
1.1 Automorphic Forms on Classical Groups
5(3)
1.2 ρ-Adic Interpolation of Automorphic Forms
8(4)
1.3 ρ-Adic Automorphic L-functions
12(1)
1.4 Galois Representations
13(1)
1.5 Plan of the Book
13(2)
1.6 Notation
15(2)
2 Geometric Reciprocity Laws 17(50)
2.1 Sketch of Classical Reciprocity Laws
18(6)
2.1.1 Quadratic Reciprocity Law
18(1)
2.1.2 Cyclotomic Version
19(1)
2.1.3 Geometric Interpretation
20(1)
2.1.4 Kronecker's Reciprocity Law
21(3)
2.1.5 Reciprocity Law for Elliptic Curves
24(1)
2.2 Cyclotomic Reciprocity Laws and Adeles
24(7)
2.2.1 Cyclotomic Fields
24(2)
2.2.2 Cyclotomic Reciprocity Laws
26(2)
2.2.3 Adelic Reformulation
28(3)
2.3 A Generalization of Galois Theory
31(5)
2.3.1 Infinite Galois Extensions
31(4)
2.3.2 Automorphism Group of a Field
35(1)
2.4 Algebraic Curves over a Field
36(15)
2.4.1 Algebraic Function Fields
36(7)
2.4.2 Zariski Topology
43(1)
2.4.3 Divisors
44(1)
2.4.4 Differentials
45(5)
2.4.5 Adele Rings of Algebraic Function Fields
50(1)
2.5 Elliptic Curves over a Field
51(11)
2.5.1 Dimension Formulas
51(1)
2.5.2 Weierstrass Equations of Elliptic Curves
52(2)
2.5.3 Moduli of Weierstrass Type
54(1)
2.5.4 Group Structure on Elliptic Curves
55(1)
2.5.5 Abel's Theorem
56(1)
2.5.6 Torsion Points on Elliptic Curves
57(3)
2.5.7 Classical Weierstrass Theory
60(2)
2.6 Elliptic Modular Function Field
62(5)
3 Modular Curves 67(30)
3.1 Basics of Elliptic Curves over a Scheme
67(5)
3.1.1 Definition of Elliptic Curves
68(1)
3.1.2 Cartier Divisors
68(1)
3.1.3 Picard Schemes
69(1)
3.1.4 Invariant Differentials
70(1)
3.1.5 Classification Functors
70(1)
3.1.6 Cartier Duality
71(1)
3.2 Moduli of Elliptic Curves and the Igusa Tower
72(14)
3.2.1 Moduli of Level 1 over Z[ 1/6]
72(2)
3.2.2 Moduli of Rhor1 (N)
74(1)
3.2.3 Action of Gm
75(2)
3.2.4 Compactification
77(2)
3.2.5 Moduli of Γ(N)-Level Structure
79(1)
3.2.6 Hasse Invariant
80(1)
3.2.7 Igusa Curves
81(1)
3.2.8 Irreducibility of Igusa Curves
82(3)
3.2.9 ρ-Adic Elliptic Modular Forms
85(1)
3.3 p-Ordinary Elliptic Modular Forms
86(6)
3.3.1 Axiomatic Treatment
86(3)
3.3.2 Bounding the p-Ordinary Rank
89(1)
3.3.3 p-Ordinary Projector
90(1)
3.3.4 Families of p-Ordinary Modular Forms
90(2)
3.4 Elliptic &Lamba;-Adic Forms and ρ-Adic L-functions
92(5)
3.4.1 Generality of &Lamba;-Adic Forms
92(2)
3.4.2 Some ρ-Adic L-Functions
94(3)
4 Hilbert Modular Varieties 97(128)
4.1 Hilbert- Blumenthal Moduli
98(33)
4.1.1 Abelian Variety with Real Multiplication
98(4)
4.1.2 Moduli Problems with Level Structure
102(2)
4.1.3 Complex Analytic Hilbert Modular Forms
104(6)
4.1.4 Toroidal Compactification
110(5)
4.1.5 Tate Semi-Abelian Schemes with Real Multiplication
115(2)
4.1.6 Hasse Invariant and Sheaves of Cusp Forms
117(2)
4.1.7 ρ-Adic Hilbert Modular Forms of Level Γ(N)
119(4)
4.1.8 Moduli Problem of Γ1/1(N)-Type
123(2)
4.1.9 ρ-Adic Modular Forms on PGL(2)
125(2)
4.1.10 Hecke Operators on Geometric Modular Forms
127(4)
4.2 Hilbert Modular Shimura Varieties
131(61)
4.2.1 Abelian Varieties up to Isogenies
133(7)
4.2.2 Global Reciprocity Law
140(14)
4.2.3 Local Reciprocity Law
154(4)
4.2.4 Hilbert Modular Igusa Towers
158(6)
4.2.5 Hecke Operators as Algebraic Correspondences
164(1)
4.2.6 Modular Line Bundles
165(10)
4.2.7 Sheaves over the Shimura Variety of PGL(2)
175(2)
4.2.8 Hecke Algebra of Finite Level
177(1)
4.2.9 Effect on q-Expansion
178(6)
4.2.10 Adelic q-Expansion
184(5)
4.2.11 Nearly Ordinary Hecke Algebra with Central Character
189(2)
4.2.12 ρ-Adic Universal Hecke Algebra
191(1)
4.3 Rank of ρ-Ordinary Cohomology Groups
192(17)
4.3.1 Archimedean Automorphic Forms
192(6)
4.3.2 Jacquet-Langlands-Shimizu Correspondence
198(4)
4.3.3 Integral Correspondence
202(3)
4.3.4 Eichler-Shimura Isomorphisms
205(1)
4.3.5 Constant Dimensionality
206(3)
4.4 Appendix: Fundamental Groups
209(16)
4.4.1 Categorical Galois Theory
209(7)
4.4.2 Algebraic Fundamental Groups
216(2)
4.4.3 Group-Theoretic Results
218(7)
5 Generalized Eichler-Shimura Map 225(26)
5.1 Semi-Simplicity of Hecke Algebras
225(11)
5.1.1 Jacquet Modules
226(1)
5.1.2 Double Coset Algebras
227(3)
5.1.3 Rational Representations of G
230(2)
5.1.4 Nearly ρ-Ordinary Representations
232(2)
5.1.5 Semi-Simplicity of Interior Cohomology Groups
234(2)
5.2 Explicit Symmetric Domains
236(8)
5.2.1 Hermitian Forms over C
236(2)
5.2.2 Symmetric Spaces of Unitary Groups
238(5)
5.2.3 Invariant Measure
243(1)
5.3 The Eichler-Shimura Map
244(7)
5.3.1 Unitary Groups
245(2)
5.3.2 Symplectic Groups
247(1)
5.3.3 Hecke Equivariance
248(3)
6 Moduli Schemes 251(52)
6.1 Hilbert Schemes
251(18)
6.1.1 Vector Bundles
252(1)
6.1.2 Grassmannians
253(3)
6.1.3 Flag Varieties
256(2)
6.1.4 Flat Quotient Modules
258(4)
6.1.5 Morphisms Between Schemes
262(2)
6.1.6 Abelian Schemes
264(5)
6.2 Quotients by PGL(η)
269(7)
6.2.1 Line Bundles on Projective Spaces
270(1)
6.2.2 Automorphism Group of a Projective Space
270(1)
6.2.3 Quotient of a Product of Projective Spaces
271(5)
6.3 Mumford Moduli
276(14)
6.3.1 Dual Abelian Scheme and Polarization
276(1)
6.3.2 Moduli Problem
276(2)
6.3.3 Abelian Scheme with Linear Rigidification
278(1)
6.3.4 Embedding into the Hilbert Scheme
279(1)
6.3.5 Conclusion
280(2)
6.3.6 Smooth Toroidal Compactification
282(8)
6.4 Siegel Modular Variety
290(13)
6.4.1 Moduli Functors
291(2)
6.4.2 Siegel Modular Reciprocity Law
293(3)
6.4.3 Siegel Modular Igusa Tower
296(7)
7 Shimura Varieties 303(26)
7.1 PEL Moduli Varieties
304(17)
7.1.1 Polarization, Endomorphism, and Lattice
304(5)
7.1.2 Construction of the Moduli
309(5)
7.1.3 Moduli Variety for Similitude Groups
314(2)
7.1.4 Classification of G
316(1)
7.1.5 Generic Fiber of Sh(&rho)/Kappa
317(4)
7.2 General Shimura Varieties
321(8)
7.2.1 Axioms Defining Shimura Varieties
321(3)
7.2.2 Reciprocity Law at Special Points
324(2)
7.2.3 Shimura's Reciprocity Law
326(3)
8 Ordinary p-Adic Automorphic Forms 329(46)
8.1 True and False Automorphic Forms
329(16)
8.1.1 An Axiomatic Igusa Tower
330(1)
8.1.2 Rational Representation and Vector Bundles
331(2)
8.1.3 Weight of Automorphic Forms and Representations
333(2)
8.1.4 Density Theorems
335(4)
8.1.5 p-Ordinary Automorphic Forms
339(2)
8.1.6 Construction of the Projector eGL
341(3)
8.1.7 Axiomatic Control Result
344(1)
8.2 Deformation Theory of Serre and Tate
345(10)
8.2.1 A Theorem of Drinfeld
346(2)
8.2.2 A Theorem of Serre-Tate
348(1)
8.2.3 Deformation of an Ordinary Abelian Variety
349(1)
8.2.4 Symplectic Case
350(1)
8.2.5 Unitary Case
351(4)
8.3 Vertical Control Theorem
355(8)
8.3.1 Hecke Operators on Deformation Space
356(4)
8.3.2 Statements and Proof
360(3)
8.4 Irreducibility of Igusa Towers
363(12)
8.4.1 Irreducibility and ρ-Decomposition Groups
364(1)
8.4.2 Closed Immersion into the Siegel Modular Variety
364(3)
8.4.3 Description of a ρ-Decomposition Group
367(2)
8.4.4 Irreducibility Theorem in Cases A and C
369(6)
References 375(8)
Symbol Index 383(2)
Statement Index 385(2)
Subject Index 387