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El. knyga: Elliptic Curves and Arithmetic Invariants

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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including µ-invariant, L-invariant, and similar topics. This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties. Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader. Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory. Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

This introduction to Shimura varieties covers key topics including non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; elliptic and modular curves over rings and more.

Recenzijos

The main aim of the book is to give an account of Hidas results on arithmetic invariants in an accessible way. The book is intended for mathematicians with some background on modular forms and is worthwhile for both graduate students and experts. There are numerous examples, exercises, and remarks, all aimed at carefully helping the reader. In conclusion, this book is a very welcome addition to the mathematical literature. (Florian Sprung, Mathematical Reviews, April, 2015)

The author gives in this book a detailed account of results concerning arithmetic invariants, including µ-invariant and L-invariant. it contains a detailed account of the authors recent results concerning arithmetic invariants. The book, addressed to advanced graduate students and experts working in number theory and arithmetic geometry, is a welcome addition to this beautiful and difficult area of research. (Andrzej Dbrowski, zbMATH, Vol. 1284, 2014)

1 Nontriviality of Arithmetic Invariants
1(42)
1.1 Arithmetic Invariants in Iwasawa Theory
3(7)
1.1.1 Iwasawa Invariant
4(1)
1.1.2 L-Invariant
5(5)
1.2 Dirichlet L-Values
10(11)
1.2.1 A Nonvanishing Result for Dirichlet L-Values
11(2)
1.2.2 Hecke Operators for Gm
13(2)
1.2.3 Measure Associated with a U(l)-Eigenfunction
15(2)
1.2.4 Evaluation Formula
17(1)
1.2.5 Zariski Density
18(2)
1.2.6 Proof of Theorem 1.8
20(1)
1.3 CM Periods and L-Values
21(22)
1.3.1 Elliptic Modular Forms
23(4)
1.3.2 A Rationality Theorem, an Application
27(2)
1.3.3 p-Adic Elliptic Modular Form
29(2)
1.3.4 CM Elliptic Curve
31(2)
1.3.5 Invariant Differential Operators
33(2)
1.3.6 p-Adic Differential Operators
35(4)
1.3.7 Katz Measure at a Glance
39(4)
2 Elliptic Curves and Modular Forms
43(40)
2.1 Curves over a Field
43(15)
2.1.1 Plane Curves
43(4)
2.1.2 Tangent Space and Local Rings
47(4)
2.1.3 Projective Space
51(1)
2.1.4 Projective Plane Curve
52(2)
2.1.5 Divisors
54(2)
2.1.6 Riemann---Roch Theorem
56(2)
2.1.7 Regular Maps from a Curve into a Projective Space
58(1)
2.2 Elliptic Curves
58(6)
2.2.1 Abel's Theorem
58(2)
2.2.2 Weierstrass Equations of Elliptic Curves
60(2)
2.2.3 Moduli of Weierstrass Type
62(2)
2.3 Modular Forms
64(10)
2.3.1 Elliptic Curves over General Rings
65(3)
2.3.2 Geometric Modular Forms
68(1)
2.3.3 Archimedean Uniformization
69(2)
2.3.4 Weierstrass Function
71(2)
2.3.5 Holomorphic Modular Forms
73(1)
2.4 p-Adic Uniformization
74(7)
2.4.1 Explicit q-Expansion
75(2)
2.4.2 Tate Curves
77(4)
2.5 What We Study in This Book
81(2)
3 Invariants, Shimura Variety, and Hecke Algebra
83(62)
3.1 Abelian Component of the "Big" Hecke Algebra
84(25)
3.1.1 Is Characterizing Abelian Components Important?
89(2)
3.1.2 Horizontal Theorem
91(1)
3.1.3 Weil Numbers
92(10)
3.1.4 A Rigidity Lemma
102(1)
3.1.5 CM Components
103(1)
3.1.6 An Eigenvalue Formula
104(2)
3.1.7 Polynomial Representations of GL(2)
106(1)
3.1.8 Proof of Theorem 3.2
107(2)
3.2 Finiteness of Abelian Varieties
109(7)
3.2.1 CM Type
112(2)
3.2.2 Supercuspidality Implies Supersingularity
114(2)
3.2.3 Twist Classes of Abelian Varieties of GL(2)-Type
116(1)
3.3 Vertical Version
116(3)
3.3.1 Results Toward the Vertical Theorem
118(1)
3.3.2 Proof of Theorem 3.25
119(1)
3.4 Nonconstancy of Adjoint L-Invariant
119(11)
3.4.1 Proof of Theorem 3.28
121(1)
3.4.2 Review of L-Invariants
122(2)
3.4.3 Galois Deformation
124(2)
3.4.4 Adjoint Selmer Groups
126(2)
3.4.5 Greenberg's £-Invariant
128(1)
3.4.6 Proof of Theorem 3.31
128(1)
3.4.7 II-adic £-Invariant
129(1)
3.5 Vanishing of the μ-Invariant of Katz L-Functions
130(9)
3.5.1 Eisenstein Series
132(3)
3.5.2 Modular Curves as Shimura Variety
135(3)
3.5.3 Subvarieties Stable under Hecke Action
138(1)
3.5.4 Conclusion
138(1)
3.6 Hecke Stable Subvariety
139(6)
3.6.1 Hecke Invariant Subvariety Is a Shimura Subvariety
140(1)
3.6.2 Rigidity Lemma and a Sketch of Proofs
140(5)
4 Review of Scheme Theory
145(72)
4.1 Functorial Algebraic Geometry
146(29)
4.1.1 Affine Variety
146(1)
4.1.2 Categories
147(1)
4.1.3 Functors
148(2)
4.1.4 Affine Schemes
150(5)
4.1.5 Zariski Open Covering
155(3)
4.1.6 Zariski Sheaves
158(6)
4.1.7 Sheaf of Differential Forms
164(3)
4.1.8 Scheme and Variety
167(2)
4.1.9 Projective Schemes
169(2)
4.1.10 Grothendieck Topology on Schemes
171(2)
4.1.11 Excellent Rings and Schemes
173(2)
4.2 Fundamental Groups
175(14)
4.2.1 Galois Extensions of Infinite Degree
175(4)
4.2.2 Automorphism Group of a Field
179(1)
4.2.3 Galois Theory in Categorical Setting
180(7)
4.2.4 Algebraization of Fundamental Groups
187(2)
4.3 Group Schemes
189(16)
4.3.1 Affine Algebraic Groups
189(4)
4.3.2 Basic Diagrams
193(1)
4.3.3 Functorial Representations
194(1)
4.3.4 Duality of Hopf Algebras
195(2)
4.3.5 Duality of Finite Flat Groups
197(2)
4.3.6 Abelian Schemes
199(1)
4.3.7 Barsotti---Tate Groups
200(1)
4.3.8 Connected---etale Exact Sequence
201(2)
4.3.9 Ordinary Barsotti---Tate Group
203(2)
4.4 Completing a Scheme
205(12)
4.4.1 Formal Schemes
206(3)
4.4.2 Deformation Functors
209(1)
4.4.3 Connected Formal Groups
210(4)
4.4.4 Infinitesimal Splitting Implies Local Splitting
214(3)
5 Geometry of Variety
217(8)
5.1 Variety over a Field
217(8)
5.1.1 Rational Map
217(3)
5.1.2 Zariski's Main Theorem
220(2)
5.1.3 Stein Factorization
222(1)
5.1.4 Provariety
223(2)
6 Elliptic and Modular Curves over Rings
225(56)
6.1 Basics of Elliptic Curves over a Scheme
225(12)
6.1.1 Definition of Elliptic Curves
226(2)
6.1.2 Cartier Divisors
228(1)
6.1.3 Picard Schemes
229(4)
6.1.4 Invariant Differentials Are Nowhere Vanishing
233(1)
6.1.5 Classification Functors
234(1)
6.1.6 Cartier Duality
235(2)
6.2 Moduli of Elliptic Curves
237(24)
6.2.1 Moduli of Level Z 1/6
238(4)
6.2.2 Compatible System of Tate Modules
242(2)
6.2.3 Moduli of & Γ1(N)
244(1)
6.2.4 Definition of Modular Forms
245(1)
6.2.5 Hecke Operators
246(1)
6.2.6 Moduli of Elliptic Curves with Level-Γ1(N) Structure
247(3)
6.2.7 Compactification and Modular Line Bundles
250(1)
6.2.8 q-Expansion
251(2)
6.2.9 Hecke Operators on q-Expansion
253(1)
6.2.10 Moduli of Principal Level Structure
254(2)
6.2.11 Hasse Invariant
256(3)
6.2.12 Igusa Curves
259(2)
6.3 Deformation of Ordinary Elliptic Curves
261(12)
6.3.1 A Theorem of Drinfeld
261(2)
6.3.2 A Theorem of Serre---Tate
263(1)
6.3.3 Group Structure on Deformation Space
264(6)
6.3.4 Computation of Deformation Coordinates
270(3)
6.4 Elliptic Curves with Complex Multiplication
273(8)
6.4.1 CM Elliptic Curves
273(2)
6.4.2 Order of an Imaginary Quadratic Field
275(1)
6.4.3 Algebraic Differential on CM Elliptic Curves
275(1)
6.4.4 CM Level Structure
276(1)
6.4.5 Adelic CM Level Structure
277(1)
6.4.6 Zeta Function of CM Elliptic Curves
278(3)
7 Modular Curves as Shimura Variety
281(54)
7.1 Shimura Curve
281(26)
7.1.1 Elliptic Modular Function Fields
282(3)
7.1.2 Complex Points of Shimura Curve
285(4)
7.1.3 Elliptic Curves Up to Isogenies
289(6)
7.1.4 Integral Shimura Curve
295(3)
7.1.5 Finite-Level Structure
298(2)
7.1.6 Adelic Action on Shimura Curves
300(1)
7.1.7 Isogeny Action
301(2)
7.1.8 Reciprocity Law at CM Points
303(3)
7.1.9 Degeneracy Operators
306(1)
7.2 Igusa Tower
307(17)
7.2.1 Axiomatic Approach to Irreducibility
307(3)
7.2.2 Mod-p Connected Components
310(4)
7.2.3 Reciprocity Law and Irreducibility of Igusa Tower
314(1)
7.2.4 p-Adic Elliptic Modular Forms
314(3)
7.2.5 Reciprocity Law for Deformation Space
317(4)
7.2.6 Katz Differential Operator
321(3)
7.3 Elliptic Modular Forms of Slope 0
324(7)
7.3.1 Control Theorems of p-Adic Modular Forms
325(3)
7.3.2 Bounding the p-Ordinary Rank
328(1)
7.3.3 Vertical Control Theorem
329(1)
7.3.4 Families of p-Ordinary Modular Forms
330(1)
7.4 Hecke Algebras
331(4)
7.4.1 Hecke Operators on p-Adic Modular Forms
331(1)
7.4.2 Hecke Operators on A-Adic Modular Forms
332(1)
7.4.3 Duality of Hecke Algebra and Hecke Modules
333(2)
8 Nonvanishing Modulo p of Hecke L-Values
335(32)
8.1 Rationality of Hecke L-Values
336(15)
8.1.1 Differential Operators and Rationality
336(1)
8.1.2 Arithmetic Hecke Characters
337(2)
8.1.3 Optimal Eisenstein Series
339(2)
8.1.4 Hecke Eigenvalues
341(4)
8.1.5 L-Functions of an Order
345(3)
8.1.6 Anticyclotomic Hecke L-Values
348(1)
8.1.7 Values of Eisenstein Series at CM Points
348(3)
8.2 Nonvanishing Modulo p of L-Values
351(16)
8.2.1 Construction of a Modular Measure
352(3)
8.2.2 Nontriviality of the Modular Measure
355(3)
8.2.3 Preliminary to the Proof of Theorem 8.25
358(2)
8.2.4 Proof of Theorem 8.25
360(1)
8.2.5 Linear Independence
361(3)
8.2.6 C-Adic Eisenstein Measure Modulo p
364(3)
9 p-Adic Hecke L-Functions and Their μ-Invariants
367(20)
9.1 Eisenstein and Katz Measure
367(10)
9.1.1 q-Expansion of Eisenstein Series
367(3)
9.1.2 Eisenstein Measure
370(1)
9.1.3 Katz Measure
371(2)
9.1.4 Evaluation of Katz Measure
373(4)
9.2 Computation of the μ-Invariant
377(10)
9.2.1 Splitting the Katz Measure
377(2)
9.2.2 Good Representatives
379(2)
9.2.3 Operators on p-Adic Modular Forms
381(1)
9.2.4 Linear Independence of Eisenstein Series
382(5)
10 Toric Subschemes in a Split Formal Torus
387(18)
10.1 Rigidity of Formal Tori
387(9)
10.1.1 Power Series Ring over a Field
388(4)
10.1.2 Linearity of Subschemes of a Formal Torus
392(4)
10.2 Representation of Lie Algebras
396(9)
10.2.1 Algebras
397(1)
10.2.2 Modules over Lie Algebras
398(1)
10.2.3 Semisimple Algebras
399(2)
10.2.4 Differential of Group Representations
401(4)
11 Hecke Stable Subvariety Is a Shimura Subvariety
405(22)
11.1 Stability under Hecke Correspondence
405(22)
11.1.1 Locally Linear Subvariety
405(1)
11.1.2 An Example of Hecke Stable Subvariety
406(2)
11.1.3 Stability under Toric Action, Local Study
408(4)
11.1.4 Stability under Toric Action, Global Study
412(12)
11.1.5 Linear Independence Again
424(3)
References 427(10)
Symbol Index 437(4)
Statement Index 441(2)
Subject Index 443
Haruzo Hida is currently a professor of mathematics at University of California, Los Angeles.
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