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El. knyga: Geometric Modular Forms And Elliptic Curves (2nd Edition)

(Univ Of California, Los Angeles, Usa)
  • Formatas: 468 pages
  • Išleidimo metai: 28-Dec-2011
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789814405232
Kitos knygos pagal šią temą:
Geometric Modular Forms And Elliptic Curves (2nd Edition)Didesnis paveikslėlis
  • Formatas: 468 pages
  • Išleidimo metai: 28-Dec-2011
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789814405232
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Hida (University of California, Los Angeles) has arranged material in five chapters covering an algebra-geometric tool box, elliptic curves, geometric modular forms, Jacobians and Galois representations, and modularity problems. New in this second edition: a detailed description of Barsotti-Tate groups, and of formal deformation theory of elliptic curves, Ribet's theorem of full image of modular p-adic Galois representation (a new result of the author), and some present day research in the area of number theory. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)

This book provides a comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura–Taniyama conjecture, is given. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction.In this new second edition, a detailed description of Barsotti–Tate groups (including formal Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to 'big' ?-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian Q-varieties and Q-curves).
Preface to the second edition v
Preface vii
1 An Algebro-Geometric Tool Box
1(104)
1.1 Sheaves
1(4)
1.1.1 Sheaves and Presheaves
1(2)
1.1.2 Sheafication
3(1)
1.1.3 Sheaf Kernel and Cokernel
4(1)
1.2 Schemes
5(8)
1.2.1 Local Ringed Spaces
5(3)
1.2.2 Schemes as Local Ringed Spaces
8(1)
1.2.3 Sheaves over Schemes
9(2)
1.2.4 Topological Properties of Schemes
11(2)
1.3 Projective Schemes
13(7)
1.3.1 Graded Rings
13(1)
1.3.2 Functor Proj
13(3)
1.3.3 Sheaves on Projective Schemes
16(4)
1.4 Categories and Functors
20(8)
1.4.1 Categories
20(2)
1.4.2 Functors
22(1)
1.4.3 Schemes as Functors
23(3)
1.4.4 Abelian Categories
26(2)
1.5 Applications of the Key-Lemma
28(10)
1.5.1 Sheaf of Differential Forms on Schemes
29(3)
1.5.2 Fiber Products
32(1)
1.5.3 Inverse Image of Sheaves
33(2)
1.5.4 Affine Schemes
35(2)
1.5.5 Morphisms into a Projective Space
37(1)
1.6 Group Schemes
38(7)
1.6.1 Group Schemes as Functors
38(1)
1.6.2 Kernel and Cokernel
39(1)
1.6.3 Bialgebras
40(2)
1.6.4 Locally Free Groups
42(2)
1.6.5 Schematic Representations
44(1)
1.7 Cartier Duality
45(5)
1.7.1 Duality of Bialgebras
45(2)
1.7.2 Duality of Locally Free Groups
47(3)
1.8 Quotients by a Group Scheme
50(12)
1.8.1 Naive Quotients
50(2)
1.8.2 Categorical Quotients
52(2)
1.8.3 Geometric Quotients
54(8)
1.9 Morphisms
62(11)
1.9.1 Topological Definitions
62(5)
1.9.2 Diffeo-Geometric Definitions
67(2)
1.9.3 Applications
69(4)
1.10 Cohomology of Coherent Sheaves
73(9)
1.10.1 Coherent Cohomology
73(4)
1.10.2 Summary of Known Facts
77(1)
1.10.3 Cohomological Dimension
78(4)
1.11 Descent
82(6)
1.11.1 Covering Data
82(1)
1.11.2 Descent Data
83(2)
1.11.3 Descent of Schemes
85(3)
1.12 Barsotti-Tate Groups
88(7)
1.12.1 p-Divisible Abelian Sheaf
88(4)
1.12.2 Connected-Etale Exact Sequence
92(1)
1.12.3 Ordinary Barsotti-Tate Group
93(2)
1.13 Formal Scheme
95(10)
1.13.1 Open Subschemes as Functors
96(1)
1.13.2 Examples of Formal Schemes
97(4)
1.13.3 Deformation Functors
101(1)
1.13.4 Connected Formal Groups
102(3)
2 Elliptic Curves
105(118)
2.1 Curves and Divisors
105(17)
2.1.1 Cartier Divisors
105(3)
2.1.2 Serre-Grothendieck Duality
108(6)
2.1.3 Riemann-Roch Theorem
114(5)
2.1.4 Relative Riemann-Roch Theorem
119(3)
2.2 Elliptic Curves
122(12)
2.2.1 Definition
122(1)
2.2.2 Abel's Theorem
123(2)
2.2.3 Holomorphic Differentials
125(1)
2.2.4 Taylor Expansion of Differentials
126(1)
2.2.5 Weierstrass Equations of Elliptic Curves
127(3)
2.2.6 Moduli of Weierstrass Type
130(4)
2.3 Geometric Modular Forms of Level 1
134(5)
2.3.1 Functorial Definition
134(2)
2.3.2 Coarse Moduli Scheme
136(2)
2.3.3 Fields of Moduli
138(1)
2.4 Elliptic Curves over C
139(6)
2.4.1 Topological Fundamental Groups
140(2)
2.4.2 Classical Weierstrass Theory
142(1)
2.4.3 Complex Modular Forms
143(2)
2.5 Elliptic Curves over p-Adic Fields
145(10)
2.5.1 Power Series Identities
145(3)
2.5.2 Universal Tate Curves
148(5)
2.5.3 Etale Covering of Tate Curves
153(2)
2.6 Level Structures
155(18)
2.6.1 Isogenics
155(2)
2.6.2 Level N Moduli Problems
157(6)
2.6.3 Generality of Elliptic Curves
163(2)
2.6.4 Proof of Theorem 2.6.8
165(3)
2.6.5 Geometric Modular Forms of Level N
168(5)
2.7 L-Functions of Elliptic Curves
173(7)
2.7.1 L-Functions over Finite Fields
173(3)
2.7.2 Hasse-Weil L-Function
176(4)
2.8 Regularity
180(9)
2.8.1 Regular Rings
180(3)
2.8.2 Regular Moduli Varieties
183(6)
2.9 p-Ordinary Moduli Problems
189(20)
2.9.1 The Hasse Invariant
189(4)
2.9.2 Ordinary Moduli of p-Power Level
193(2)
2.9.3 Irreducibility of p-Ordinary Moduli
195(1)
2.9.4 Moduli Problem of Γ0 and Γ1 Type
196(2)
2.9.5 Moduli Problem of Γ0(p) and Γ1(p) Type
198(11)
2.10 Deformation of Elliptic Curves
209(14)
2.10.1 A Theorem of Drinfeld
209(2)
2.10.2 A Theorem of Serre-Tate
211(3)
2.10.3 Deformation of an Ordinary Elliptic Curve
214(9)
3 Geometric Modular Forms
223(64)
3.1 Integrality
223(15)
3.1.1 Spaces of Modular Forms
223(13)
3.1.2 Horizontal Control Theorem
236(2)
3.2 Vertical Control Theorem
238(38)
3.2.1 False Modular Forms
240(12)
3.2.2 p-Adic Modular Forms
252(5)
3.2.3 Hecke Operators
257(9)
3.2.4 Families of p-Adic Modular Forms
266(5)
3.2.5 Horizontal Control of p-Power Level
271(2)
3.2.6 Control of Hecke algebra
273(2)
3.2.7 Irreducible Components and Analytic Families
275(1)
3.3 Action of GL(2) on Modular Forms
276(11)
3.3.1 Action of GL2(Z/NZ)
276(4)
3.3.2 Action of GL2(Z)
280(7)
4 Jacobians and Galois Representations
287(96)
4.1 Jacobians of Stable Curves
287(35)
4.1.1 Non-Singular Curves
287(8)
4.1.2 Union of Two Curves
295(3)
4.1.3 Functorial Properties of Jacobians
298(4)
4.1.4 Self-Duality of Jacobian Schemes
302(2)
4.1.5 Generality on Abelian Schemes
304(9)
4.1.6 Endomorphism of Abelian Schemes
313(5)
4.1.7 l-Adic Galois Representations
318(4)
4.2 Modular Galois Representations
322(20)
4.2.1 Hecke Correspondences
323(3)
4.2.2 Galois Representations on Modular Jacobians
326(4)
4.2.3 Ramification at the Level
330(5)
4.2.4 Ramification of p-Adic Representations at p
335(2)
4.2.5 Modular Galois Representations of Higher Weight
337(5)
4.3 Fullness of Big Galois Representations
342(41)
4.3.1 Big I-adic Galois Representations
344(1)
4.3.2 Ramification of I-adic Galois Representations
345(1)
4.3.3 Lie Algebras over p-Adic Ring
346(2)
4.3.4 Lie Algebras of p-Profinite Subgroups of SL(2)
348(7)
4.3.5 Lie Algebra and Lie Group over Zp
355(4)
4.3.6 Arithmetic Galois Characters
359(2)
4.3.7 Fullness of Modular Galois Representation
361(4)
4.3.8 Fullness of Elliptic Curves
365(3)
4.3.9 Fullness of Lie Algebra over Λ
368(3)
4.3.10 Fullness of I-Adic Galois Representation
371(2)
4.3.11 Basic Subgroups
373(7)
4.3.12 Proof of Theorem 4.3.4
380(3)
5 Modularity Problems
383(54)
5.1 Induced and Extended Galois Representations
384(18)
5.1.1 Induction and Extension
385(7)
5.1.2 Automorphic Induction
392(3)
5.1.3 Artin Representations
395(7)
5.2 Some Other Solutions
402(14)
5.2.1 A Theorem of Wiles
402(2)
5.2.2 Modularity of Extended Galois Representations
404(2)
5.2.3 Elliptic Q-Curves
406(7)
5.2.4 Shimura-Taniyama Conjecture
413(3)
5.3 Modularity of Abelian Q-Varieties
416(21)
5.3.1 Abelian F-varieties of GL(2)-type
417(7)
5.3.2 Endomorphism Algebras of Abelian F-varieties
424(1)
5.3.3 Application to Abelian Q-Varieties
425(7)
5.3.4 Abelian Varieties with Real Multiplication
432(5)
Bibliography 437(10)
List of Symbols 447(2)
Statement Index 449(2)
Index 451
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