Arithmetic Moduli Of Elliptic Curves

, by Katz, Nicholas M.

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ISBN: 9780691083520 | 0691083525
Cover: Paperback
Copyright: 2/1/1985

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Supplemental Materials

Introduction

Generalities on ``A-Structures'' and ``A-Generators''

(60)

Review of relative Cartier divisors

(4)

Relative Cartier divisors in curves

(5)

Existence of incidence schemes

(5)

Points of ``exact order N'' and cyclic subgroups

(3)

A mild generalization: A -structures and A -generators

(2)

General representability theorems for A -structures and A -generators

(4)

Factorization into prime powers of A -structures and A -generators

(6)

Full sets of sections

(6)

Intrinsic A -structures and A -generators

(2)

Relation to Cartier divisors

(8)

Extensions of an etale group

(7)

Roots of unity

(6)

Some open problems

(2)

Review of Elliptic Curves

(35)

The group structure

(4)

Generalized Weierstrass equations, and some elementary universal families

(6)

The structure of [N]

(2)

Rigidity

(2)

Manifestations of autoduality

(4)

Hasse's theory

(4)

Applications to rigidity

(2)

Pairings

(4)

Deformation theory

(7)

The Four Basic Moduli Problems for Elliptic Curves: Sorites

(9)

Γ (N)-structures

(1)

Γ1 (N)-structures

(1)

Balanced Γ1 (N)-structures

100

(1)

Γ0(N)-structures

100

(1)

Factorization into prime powers

101

(1)

Relative representability

102

(2)

The situation when N is invertible

104

(3)

The Formalism of Moduli Problems

107

(18)

The category (E11)

107

(1)

Moduli problems

107

(1)

Representable moduli problems

108

(1)

Rigid moduli problems

109

(1)

Geometric properties of moduli problems

109

(1)

Some examples

110

(1)

A basic result: representability and rigidity

111

(6)

Another example

117

(1)

Yet another example

117

(1)

Lemmas on group-schemes

118

(2)

Modular families

120

(1)

More geometric properties of moduli problems

121

(1)

The category (E11/R)

122

(1)

Moduli problems of finite level

123

(2)

APPENDIX: MORE ON RIGIDITY AND REPRESENTABILITY

125

(90)

Regularity theorems

129

(23)

First main theorem

129

(1)

Axiomatics

129

(6)

End of the proof

135

(8)

Summary of parameters at supersingular points

143

(1)

First applications

143

(7)

Pairings

150

(2)

Cyclicity

152

(34)

The main theorem

152

(3)

Axiomatics

155

(1)

End of the proof

156

(6)

Cyclicity as a closed condition

162

(2)

The moduli problem [N-Isog]

164

(2)

The moduli problem [Γ0(N)]: proof of the First Main Theorem

166

(1)

Detailed theory of cyclic isogenies and cyclic subgroups; standard factorizations

167

(11)

More on [N-Isog]

178

(8)

Quotients by Finite Groups

186

(29)

The general situation

186

(9)

A descent situation

195

(1)

Quotients of product problems

196

(1)

Applications to the four basic moduli problems

197

(4)

Axiomatics

201

(6)

Applications to regularity

207

(1)

Summary of parameters at supersingular points

208

(1)

More parameters for [Γ0(pn)] at supersingular points

208

(2)

Detailed study of the congruence quotients [Γ0(pn); a, b] of [bal. Γ1(pn)]

210

(5)

APPENDIX: BASE CHANGE FOR RINGS OF INVARIANTS

215

(290)

Coarse Moduli Schemes, Cusps, and Compactifications

224

(47)

Coarse moduli schemes

224

(4)

The j-line as a coarse moduli scheme

228

(4)

Localization of moduli problems over the j-line

232

(2)

The j-invariant as a line modulus, coarse moduli schemes as fine moduli schemes (!)

234

(9)

Base change for coarse moduli schemes

243

(3)

Cusps by normalization near infinity; compactified coarse moduli schemes

246

(5)

Interlude: The groups T[N] and T

251

(7)

Relation to the Tate curve

258

(2)

Relation with ordinary elliptic curves via the Serre-Tate parameter

260

(1)

Other universality properties of the groups T[N]

261

(5)

Computation of Cusps(P) via the Tate curve

266

(5)

Moduli Problems Viewed Over Cyclotomic Integer Rings

271

(15)

Generalities

271

(6)

A descent situation

277

(1)

The situation near infinity

278

(3)

Applications to the basic moduli problems

281

(5)

The Calculus of Cusps and Components Via the Groups T[N] and the Global Structure of the Basic Moduli Problems

286

(53)

Motivation

286

(1)

Analysis of [Γ(N)]

287

(3)

Group action

290

(3)

Canonical problems

293

(2)

Explication for T[N]

295

(1)

Cusp-labels and component-labels

295

(1)

Some combinatorial lemmas

296

(3)

Application to structure near infinity

299

(2)

Applications to the four basic moduli problems

301

(8)

Detailed analysis at a prime p, balanced subgroups

309

(15)

Basic examples of balanced subgroups

324

(2)

Application to the moduli problem [Γ(pn); a, a]

326

(2)

The numerology of moduli schemes, via the line bundle ω

328

(11)

Interlude: Exotic Modular Morphisms and Isomorphisms

339

(5)

Motivation

339

(1)

``Abstract'' morphisms

339

(1)

Some basic examples

340

(4)

New Moduli Problems in Characteristic p; Igusa Curves

344

(45)

Frobenius

344

(1)

Basic lemmas

345

(4)

Igusa structures

349

(4)

The Hasse invariant

353

(6)

Ordinary curves

359

(2)

First analysis of the Igusa curve

361

(5)

Analysis of cusps

366

(2)

The equation defining Ig(p) , and a theorem of Serre

368

(8)

Numerology of Igusa curves

376

(5)

``Exotic'' projections from Igusa curves; ``exotic'' Igusa structures

381

(8)

Reductions mod p of the Basic Moduli Problems

389

(68)

Some general considerations on crossings at supersingular points

389

(7)

Modular schemes as examples

396

(3)

Analysis of p-power isogenies between elliptic curves

399

(8)

Global structure of the moduli spaces M(P, [Gamma;0(pn)], M(P, [pn-Isog])

407

(7)

Analysis of [Γ1(PN)] in characteristic p

414

(4)

Explicit calculations via the groups T[pn] of [Γ0(pn)],[Γ1(pn)]

418

(6)

The reduction mod p of [Γ(pn)can

424

(5)

Complete local ring of [Γ(pn)]can at supersingular points; intersection numbers

429

(4)

Distribution of the cusps on [Γ(pn)]can

433

(2)

The reduction mod p of a general p-power level moduli problem

435

(6)

The reduction mod p of [bal. Γ1(pn)]can

441

(9)

The reduction mod p of quotients of [bal. Γ1(pn)] by subgroups of (Z/pnZ)xx(Z/pnZ)X

450

(7)

Application to Theorems of Good Reduction

457

(48)

General review of vanishing cycles

457

(10)

Application to curves

467

(4)

Application to modular curves: explicitation of the numerical criterion

471

(8)

Characters and conductors

479

(1)

The Good Reduction Theorem

480

(20)

Explicitation of the Good Reduction Theorem

500

(2)

Application to Jacobians

502

(3)

Notes Added in Proof

505

(6)

References

511

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Arithmetic Moduli Of Elliptic Curves

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