Geometric Modular Forms and Elliptic Curves

by Hida, Haruzo

• ISBN13: 9789810243371
• ISBN10: 9810243375
• eBook ISBN(s): 9789812792693
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: WORLD SCIENTIFIC PUB CO INC
• More Book Details
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Summary

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.

Table of Contents

Preface	v
An Algebro-Geometric Tool Box	1	(88)
Sheaves	1	(4)
Sheaves and Presheaves	1	(1)
Sheafication	2	(2)
Sheaf Kernel and Cokernel	4	(1)
Schemes	5	(8)
Local Ringed Spaces	5	(3)
Schemes as Local Ringed Spaces	8	(1)
Sheaves over Schemes	9	(2)
Topological Properties of Schemes	11	(2)
Projective Schemes	13	(7)
Graded Rings	13	(1)
Functor Proj	13	(4)
Sheaves on Projective Schemes	17	(3)
Categories and Functors	20	(8)
Categories	21	(1)
Functors	22	(1)
Schemes as Functors	23	(2)
Abelian Categories	25	(3)
Applications of the Key-Lemma	28	(9)
Sheaf of Differential Forms on Schemes	28	(3)
Fiber Products	31	(1)
Inverse Image of Sheaves	32	(3)
Affine Schemes	35	(1)
Morphisms into a Projective Space	36	(1)
Group Schemes	37	(8)
Group Schemes as Functors	37	(2)
Kernel and Cokernel	39	(1)
Bialgebras	40	(2)
Locally Free Groups	42	(2)
Schematic Representations	44	(1)
Cartier Duality	45	(5)
Duality of Bialgebras	45	(2)
Duality of Locally Free Groups	47	(3)
Quotients by a Group Scheme	50	(12)
Naive Quotients	50	(2)
Categorical Quotients	52	(2)
Geometric Quotients	54	(8)
Morphisms	62	(10)
Topological Definitions	62	(5)
Diffeo-Geometric Definitions	67	(2)
Applications	69	(3)
Cohomology of Coherent Sheaves	72	(10)
Coherent Cohomology	73	(4)
Summary of Known Facts	77	(2)
Cohomological Dimension	79	(3)
Descent	82	(7)
Covering Data	82	(2)
Descent Data	84	(2)
An Application to Schemes	86	(3)
Elliptic Curves	89	(108)
Curves and Divisors	89	(18)
Cartier Divisors	89	(4)
Grothendieck-Serre Duality	93	(5)
Riemann-Roch Theorem	98	(5)
Relative Riemann-Roch Theorem	103	(4)
Elliptic Curves	107	(12)
Definition	107	(1)
Abel's Theorem	108	(2)
Holomorphic Differentials	110	(1)
Taylor Expansion of Differentials	111	(1)
Weierstrass Equations of Elliptic Curves	112	(3)
Moduli of Weierstrass Type	115	(4)
Geometric Modular Forms of Level 1	119	(5)
Functorial Definition	119	(1)
Coarse Moduli Scheme	120	(3)
Fields of Moduli	123	(1)
Elliptic Curves over C	124	(6)
Topological Fundamental Groups	125	(2)
Classical Weierstrass Theory	127	(1)
Complex Modular Forms	128	(2)
Elliptic Curves over p-Adic Fields	130	(10)
Power Series Identities	130	(3)
Universal Tate Curves	133	(5)
Etale Covering of Tate Curves	138	(2)
Level Structures	140	(20)
Isogenies	140	(3)
Level N Moduli Problems	143	(6)
Generality of Elliptic Curves	149	(2)
Proof of Theorem 2.6.8	151	(3)
Geometric Modular Forms of Level N	154	(6)
L-Functions of Elliptic Curves	160	(6)
L-Functions over Finite Fields	160	(2)
Hasse-Weil L-Function	162	(4)
Regularity	166	(9)
Regular Rings	167	(3)
Regular Moduli Varieties	170	(5)
p-Ordinary Moduli Problems	175	(22)
The Hasse Invariant	176	(3)
Ordinary Moduli of p-Power Level	179	(3)
Irreducibility of p-Ordinary Moduli	182	(1)
Moduli Problem of Γ0 and Γ1 Type	183	(2)
Moduli Problem of Γ0(p) and Γ1(p) Type	185	(12)
Geometric Modular Forms	197	(60)
Integrality	197	(15)
Spaces of Modular Forms	197	(13)
Horizontal Control Theorem	210	(2)
Vertical Control Theorem	212	(34)
False Modular Forms	214	(13)
p-Adic Modular Forms	227	(5)
Hecke Operators	232	(7)
Families of p-Adic Modular Forms	239	(5)
Horizontal Control of p-Power Level	244	(2)
Action of GL(2) on Modular Forms	246	(11)
Action of GL2 (Z/NZ)	246	(4)
Action of GL2(Z)	250	(7)
Jacobians and Galois Representations	257	(56)
Jacobians of Stable Curves	257	(35)
Non-Singular Curves	257	(8)
Union of Two Curves	265	(3)
Functorial Properties of Jacobians	268	(4)
Self-Duality of Jacobian Schemes	272	(2)
Generality on Abelian Schemes	274	(9)
Endomorphism of Abelian Schemes	283	(5)
ℓ-Adic Galois Representations	288	(4)
Modular Galois Representations	292	(21)
Hecke Correspondences	293	(3)
Galois Representations on Modular Jacobians	296	(4)
Ramification at the Level	300	(6)
Ramification of p-Adic Representations at p	306	(2)
Modular Galois Representations of Higher Weight	308	(5)
Modularity Problems	313	(34)
Induced and Extended Galois Representations	314	(17)
Induction and Extension	314	(8)
Automorphic Induction	322	(3)
Artin Representations	325	(6)
Some Other Solutions	331	(16)
A Theorem of Wiles	331	(2)
Modularity of Extended Galois Representations	333	(3)
Elliptic Q-Curves	336	(7)
Shimura-Taniyama Conjecture	343	(4)
Bibliography	347	(8)
List of Symbols	355	(2)
List of Statements	357	(2)
Index	359