Arithmetic algebraic geometry / Brian Conrad, Karl Rubin, editors.

Format:

Book

Contributor:

Conrad, Brian, 1970-

Rubin, Karl.

Series:

IAS/Park City mathematics series 1079-5634 ; v. 9.

IAS/Park City mathematics series, 1079-5634 ; v. 9

Language:

English

Subjects (All):

Arithmetical algebraic geometry--Congresses.

Arithmetical algebraic geometry.

Genre:

Conference papers and proceedings.

Physical Description:

xiv, 569 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society : Institute for Advanced Study, [2001]

Contents:

Elliptic Curves, Modular Forms, and Applications / Joe P. Buhler 5

Lecture 1. Elliptic Curves 9

Curves in the affine plane 11

Projective curves 14

The group operation 17

Rational maps 18

Genus 19

Weierstrass form and the group law via divisors 21

Lecture 2. Points on Elliptic Curves 25

Finding points 25

Minimal discriminants 27

Reduction 28

The p-adic filtration 29

Lecture 3. Elliptic Curves Over C 33

Fundamental isomorphisms 33

The j invariant of an elliptic curve 34

Elliptic curves via lattices 36

Lattices via the upper half-plane 38

Lecture 4. Modular Forms of Level 1 41

Weak modular forms 41

Modular Forms 42

Dimensions of spaces of modular forms 44

Hecke operators 46

Lecture 5. L-series; Modular Forms of Higher Level 49

Dirichlet series 49

L-functions of modular forms of level 1 50

Forms of higher level 51

L-series of elliptic curves 54

Lecture 6. L-adic Representations 57

Galois representations 57

One-dimensional representations 58

Galois representations coming from elliptic curves 59

The conductor 60

Lecture 7. The Rank of Elliptic Curves Over Q 63

Mordell-Weil 64

The Weak Mordell-Weil Theorem 65

The analytic rank 69

Lecture 8. Applications of Elliptic Curves 71

Primality 71

Factoring 73

Cryptography 75

Open Questions in Arithmetic Algebraic Geometry / Alice Silverberg 83

Elliptic curves 86

Abelian varieties 87

Mordell-Weil groups 88

Torsion subgroups 89

Ranks 90

Looking ahead 90

Chapter 2. Torsion Subgroups 91

Torsion subgroups for elliptic curves 91

A quick look at modular curves 92

Conjectures 93

Other results for elliptic curves 93

Abelian varieties over number fields 95

Abelian varieties over other fields 98

Digression on complex multiplication 98

Chapter 3. Ranks 99

Unboundedness/boundedness 100

Vary the field 101

Quadratic twists 101

Average ranks 103

Distribution of ranks 104

Chapter 4. Conjectures of Birch and Swinnerton-Dyer 107

The L-function of an elliptic curve 107

Modularity 109

BSD I 111

The Congruent Number Problem 112

Evidence for BSD I 112

BSD II 113

Evidence for BSD II 115

Selmer groups 116

Examples

descents 117

Chapter 5. ABC and Related Conjectures 123

ABC Conjectures 123

ABC Conjectures for function fields 125

Szpiro Conjecture 125

Goldfeld-Szpiro Conjecture 126

de Weger Conjectures 127

Additional remarks 128

Chapter 6. Some Other Conjectures 129

Integral points on elliptic curves 129

Bounds on numbers of rational points on curves 129

Conductors of elliptic curves 130

Abelian varieties, Shimura varieties, and open questions 130

Lectures on Serre's Conjectures / Kenneth A. Ribet, William A. Stein 143

Chapter 1. Introduction to Serre's Conjecture 149

The weak conjecture of Serre 153

The strong conjecture 154

Representations arising from an elliptic curve 156

Background material 157

The cyclotomic character 157

Frobenius elements 158

Modular forms 159

Tate curves 160

Mod l modular forms 161

Chapter 2. Optimizing the Weight 163

Representations arising from forms of low weight 164

The ordinary case 164

The supersingular case and fundamental characters 164

Representations of high weight 166

The supersingular case 168

Systems of mod l eigenvalues 169

The supersingular case revisited 170

The ordinary case 172

Distinguishing between weights 2 and l + 1 172

Geometric construction of Galois representations 172

Representations arising from elliptic curves 174

Frey curves 175

Companion forms 176

Chapter 3. Optimizing the Level 177

Reduction to weight 2 177

Geometric realization of Galois representations 178

Multiplicity one 179

Multiplicity one representations 180

Multiplicity one theorems 181

Multiplicity one for mod 2 representations 181

The key case 182

Approaches to level optimization in the key case 183

Some commutative algebra 184

Aside: Examples in characteristic two 184

III applies but I and II do not 185

II applies but I and III do not 185

Aside: Sketching the spectrum of the Hecke algebra 186

Mazur's principle 187

Level optimization using a pivot 190

Shimura curves 191

Character groups 191

Proof 192

Level optimization with multiplicity one 193

Solutions 200

Chapter 5. Appendix by Brian Conrad: The Shimura Construction in Weight 2 205

Analytic preparations 205

Algebraic preliminaries 212

Proof of Theorem 5.12 216

Chapter 6. Appendix by Kevin Buzzard: A Mod l Multiplicity One Result 223

Deformations of Galois Representations / Fernando Q. Gouvea 233

Lecture 1. Galois Groups and Their Representations 237

Galois groups of infinite algebraic extensions 237

The galois group of Q 241

Restricting the ramification 244

Galois representations 247

Complements to Lecture 1 249

Lecture 2. Deformations of Representations 253

Why deform Galois representations? 253

The deformation functor 255

Universal deformations: why representable functors are nice 261

Representable functors and fiber products 262

The tangent space 266

Complements to Lecture 2 268

Lecture 3. The Universal Deformation: Existence 271

Schlessinger's criteria 272

Universal Deformations exist 275

Absolutely irreducible representations 278

Example: the case n = 1 279

Complements of Lecture 3 282

Lecture 4. The Universal Deformation: Properties 283

Functorial properties 283

Tangent spaces and cohomology groups 284

Tangent spaces and extensions of modules 285

Obstructed and unobstructed deformation problems 287

Galois representations 290

Lecture 5. Explicit Deformations 293

Group theory 295

Pro-p-extensions 296

Tame representations 298

Lecture 6. Deformations With Prescribed Properties 303

Deformation conditions 304

Deformations with fixed determinant 306

Categorical deformation conditions 308

Ordinary deformations 309

Deformation conditions for global Galois representations 311

Representations that are ordinary at p 313

Complements to Lecture 6 313

Lecture 7. Modular Deformations 317

Classical modular forms and their representations 318

p-adic modular forms 323

The ordinary case 327

Imposing deformation conditions 328

Lecture 8. p-adic Families and Infinite Ferns 331

The slope of an eigenform 332

p-old and p-new 333

p-adic families of modular forms 335

Infinite ferns 335

Appendix 1. A Criterion for Existence of a Universal Deformation Ring / Mark Dickinson 339

Appendix 2. An Overview of a Theorem of Flach / Tom Weston 345

Unobstructed deformation problems 346

Galois modules and the calculus of Tate twists 346

First reductions 348

Selmer groups 351

The L-function of Sym[superscript 2] T[subscript l] 354

Kolyvagin's theory of Euler systems 356

The Flach map 360

The geometry of modular curves 363

Some modular units 366

Local behavior of the c[subscript r] 367

Appendix A On the local Galois invariants of E[l] [multiply sign in circle] E[l] 370

Appendix B The definition of the Flach map 373

Appendix 3. An Introduction to the p-adic Geometry of Modular Curves / Matthew Emerton 377

The curve X[subscript 0](2) 378

Canonical subgroups of elliptic curves over C[subscript 2] 382

The parameter q 384

The Hasse invariant 386

Return to X[subscript 0](2) 387

The theory for arbitrary primes p 390

p-adic modular forms 393

Guide to the literature 395

Introduction to Iwasawa Theory for Elliptic Curves / Ralph Greenberg 407

Chapter 1. Mordell-Weil Groups 411

Chapter 2. Selmer Groups 425

Chapter 3. A-Modules 439

Chapter 4. Mazur's Control Theorem 453

Galois Cohomology / John Tate 465

Group modules 467

Cohomology 468

Characterization of H[superscript r] (G, -) 470

Kummer

theory 470

Functor of pairs (G, A) 472

The Shafarevich group 473

The inflation-restriction sequence 474

Cup products 475

G-pairing 475

Duality for finite modules 476

Local Fields 476

The Arithmetic of Modular Forms / Wen-Ching Winnie Li 481

Lecture 1. Introduction to Elliptic Curves, Modular Forms, and Calabi-Yau Varieties 485

Classical modular forms 485

Elliptic curves 487

Zeta functions 488

Calabi-Yau varieties 489

Lecture 2. The arithmetic of modular forms 491

Eisenstein series 491

Hecke operators 492

The structure of C(k, N, x)- The theory of newforms 493

Lecture 3. Connections Among Modular Forms, Elliptic Curves, and Representations of Galois Groups 497

Functional equations 497

Connections with elliptic curves 498

Connections with representations of the Galois group over Q 500

Comparisons with automorphic forms of GL[subscript 2] over function fields 502

Arithmetic of Certain Calabi-Yau Varieties and Mirror Symmetry / Noriko Yui 507

Lecture 1. The Modularity Conjecture for Rigid Calabi-Yau Threefolds over the Field of Rational Numbers 513

Definition of Calabi-Yau varieties 513

The mirror symmetry conjecture 516

Rigid Calabi-Yau threefolds over Q 517

The modularity conjecture for rigid Calabi-Yau threefolds over Q 517

Evidence for the modularity conjecture 519

Strategy for establishing the modularity conjecture 522

The intermediate Jacobian of a Calabi-Yau threefold 524

The order of vanishing of L-series at s = 2 526

The conjecture of Beilinson and Bloch 527

Lecture 2. Arithmetic of Orbifold Calabi-Yau Varieties Over Number Fields 531

Fermat hypersurfaces and their deformations 531

Orbifold Calabi-Yau varieties 532

Sketch of proof of Theorem 2.2 535

The L-series of orbifold Calabi-Yau varieties 538

Sketch of proof of Theorem 4.2 539

The Tate conjecture 540

The conjecture of Beilinson and Bloch 541

Construction of Mirror Calabi-Yau varieties 543

Lecture 3. K3 Surfaces, Mirror Moonshine Phenomenon 549

Mirror Moonshine phenomena 549

K3 surfaces 550

The K3 lattice 552

Lattice polarized K3 surfaces 553

Moduli space of lattice polarized K3 surfaces 554

Picard-Fuchs differential equations for one-parameter families of K3 surfaces 557

Mirror maps 558

Generalizations and open problems 562.

Notes:

"Contains the lecture notes from the Graduate Summer School program on Arithmetic Algebraic Geometry held in Park City, Utah, on June 20-July 10, 1999"--T.p. verso.

Includes bibliographical references (pages 565-569).

ISBN:

0821821733

OCLC:

46785023