Introduction to modern number theory: fundamental problems, ideas and theories
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| Format: | Buch |
| Sprache: | English |
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Berlin [u.a.]
Springer
2007
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| Ausgabe: | 2. ed., 2. corr. print. |
| Schriftenreihe: | Encyclopaedia of mathematical sciences
49 Encyclopaedia of mathematical sciences Number theory ; 1 |
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| Beschreibung: | XV, 514 S. graph. Darst. |
| ISBN: | 9783540203643 |
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| 100 | 1 | |a Manin, Jurij I. |d 1937-2023 |0 (DE-588)121177580 |4 aut | |
| 245 | 1 | 0 | |a Introduction to modern number theory |b fundamental problems, ideas and theories |c Yuri Ivanovic Manin ; Alexei A. Panchishkin |
| 250 | |a 2. ed., 2. corr. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XV, 514 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Encyclopaedia of mathematical sciences |v 49 | |
| 490 | 1 | |a Encyclopaedia of mathematical sciences : Number theory |v 1 | |
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| 700 | 1 | |a Pančiškin, Aleksej A. |d 1953- |0 (DE-588)128761407 |4 aut | |
| 830 | 0 | |a Encyclopaedia of mathematical sciences |v 49 |w (DE-604)BV024126459 |9 49 | |
| 830 | 0 | |a Encyclopaedia of mathematical sciences |v Number theory ; 1 |w (DE-604)BV020831274 |9 1 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016152126&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1804137189013979136 |
|---|---|
| adam_text | Contents
Part I Problems and Tricks
Elementary Number Theory............................... 9
1.1 Problems About Primes. Divisibility and Primality.......... 9
1.1.1 Arithmetical Notation ............................. 9
1.1.2 Primes and composite numbers ..................... 10
1.1.3 The Factorization Theorem and the Euclidean
Algorithm........................................ 12
1.1.4 Calculations with Residue Classes................... 13
1.1.5 The Quadratic Reciprocity Law and Its Use .......... 15
1.1.6 The Distribution of Primes......................... 17
1.2 Diophantine Equations of Degree One and Two.............. 22
1.2.1 The Equation ax + by = c.......................... 22
1.2.2 Linear Diophantine Systems........................ 22
1.2.3 Equations of Degree Two........................... 24
1.2.4 The Minkowski-Hasse Principle for Quadratic Forms... 26
1.2.5 Pell s Equation.................................... 28
1.2.6 Representation of Integers and Quadratic Forms by
Quadratic Forms.................................. 29
1.2.7 Analytic Methods................................. 33
1.2.8 Equivalence of Binary Quadratic Forms.............. 35
1.3 Cubic Diophantine Equations............................. 38
1.3.1 The Problem of the Existence of a Solution........... 38
1.3.2 Addition of Points on a Cubic Curve................. 38
1.3.3 The Structure of the Group of Rational Points of a
Non-Singular Cubic Curve ......................... 40
1.3.4 Cubic Congruences Modulo a Prime................. 47
1.4 Approximations and Continued Fractions................... 50
1.4.1 Best Approximations to Irrational Numbers .......... 50
1.4.2 Farey Series...................................... 50
1.4.3 Continued Fractions............................... 51
VIII Contents
1.4.4 SL2-Equivalence.................................. 53
1.4.5 Periodic Continued Fractions and Pell s Equation...... 53
1.5 Diophantine Approximation and the Irrationality............ 55
1.5.1 Ideas in the Proof that ((3) is Irrational.............. 55
1.5.2 The Measure of Irrationality of a Number............ 56
1.5.3 The Thue-Siegel-Roth Theorem, Transcendental
Numbers, and Diophantine Equations................ 57
1.5.4 Proofs of the Identities (1.5.1) and (1.5.2)............ 58
1.5.5 The Recurrent Sequences an and bn ................. 59
1.5.6 Transcendental Numbers and the Seventh Hubert
Problem.......................................... 61
1.5.7 Work of Yu.V. Nesterenko on e*, [Nes99]............. 61
2 Some Applications of Elementary Number Theory......... 63
2.1 Factorization and Public Key Cryptosystems................ 63
2.1.1 Factorization is Time-Consuming.................... 63
2.1.2 One-Way Functions and Public Key Encryption....... 63
2.1.3 A Public Key Cryptosystem........................ 64
2.1.4 Statistics and Mass Production of Primes............. 66
2.1.5 Probabilistic Primality Tests........................ 66
2.1.6 The Discrete Logarithm Problem and The
Diffie-Hellman Key Exchange Protocol............... 67
2.1.7 Computing of the Discrete Logarithm on Elliptic
Curves over Finite Fields (ECDLP).................. 68
2.2 Deterministic Primality Tests............................. 69
2.2.1 Adleman-Pomerance-Rumely Primality Test: Basic
Ideas ............................................ 69
2.2.2 Gauss Sums and Their Use in Primality Testing....... 71
2.2.3 Detailed Description of the Primality Test............ 75
2.2.4 Primes is in P .................................... 78
2.2.5 The algorithm of M. Agrawal, N. Kayal and N. Saxena . 81
2.2.6 Practical and Theoretical Primality Proving. The
ECPP (Elliptic Curve Primality Proving by F.Morain,
see [AtMo93b]).................................... 81
2.2.7 Primes in Arithmetic Progression.................... 82
2.3 Factorization of Large Integers............................ 84
2.3.1 Comparative Difficulty of Primality Testing and
Factorization ..................................... 84
2.3.2 Factorization and Quadratic Forms.................. 84
2.3.3 The Probabilistic Algorithm CLASNO............... 85
2.3.4 The Continued Fractions Method (CFRAC) and Real
Quadratic Fields.................................. 87
2.3.5 The Use of Elliptic Curves.......................... 90
Contents IX
Part II Ideas and Theories
3 Induction and Recursion................................... 95
3.1 Elementary Number Theory From the Point of View of Logic . 95
3.1.1 Elementary Number Theory........................ 95
3.1.2 Logic............................................ 96
3.2 Diophantine Sets ....................................... 98
3.2.1 Enumerability and Diophantine Sets ................ 98
3.2.2 Diophantineness of enumerable sets.................. 98
3.2.3 First properties of Diophantine sets ................. 98
3.2.4 Diophantineness and Pell s Equation................. 99
3.2.5 The Graph of the Exponent is Diophantine...........100
3.2.6 Diophantineness and Binomial coefficients............100
3.2.7 Binomial coefficients as remainders..................101
3.2.8 Diophantineness of the Factorial.....................101
3.2.9 Factorial and Euclidean Division....................101
3.2.10 Supplementary Results.............................102
3.3 Partially Recursive Functions and Enumerable Sets..........103
3.3.1 Partial Functions and Computable Functions .........103
3.3.2 The Simple Functions..............................103
3.3.3 Elementary Operations on Partial functions...........103
3.3.4 Partially Recursive Description of a Function.........104
3.3.5 Other Recursive Functions..........................106
3.3.6 Further Properties of Recursive Functions............108
3.3.7 Link with Level Sets...............................108
3.3.8 Link with Projections of Level Sets..................108
3.3.9 Matiyasevich s Theorem............................109
3.3.10 The existence of certain bijections...................109
3.3.11 Operations on primitively enumerable sets............111
3.3.12 Gödel s function...................................111
3.3.13 Discussion of the Properties of Enumerable Sets.......112
3.4 Diophantineness of a Set and algorithmic Undecidability......113
3.4.1 Algorithmic undecidability and unsolvability..........113
3.4.2 Sketch Proof of the Matiyasevich Theorem ...........113
4 Arithmetic of algebraic numbers...........................115
4.1 Algebraic Numbers: Their Realizations and Geometry........115
4.1.1 Adjoining Roots of Polynomials.....................115
4.1.2 Galois Extensions and Frobenius Elements............117
4.1.3 Tensor Products of Fields and Geometrie Realizations
of Algebraic Numbers..............................119
4.1.4 Units, the Logarithmic Map, and the Regulator.......121
4.1.5 Lattice Points in a Convex Body....................123
Contents
4.1.6 Deduction of Dirichlet s Theorem From Minkowski s
Lemma ..........................................125
4.2 Decomposition of Prime Ideals, Dedekind Domains, and
Valuations..............................................126
4.2.1 Prime Ideals and the Unique Factorization Property . . . 126
4.2.2 Finiteness of the Class Number .....................128
4.2.3 Decomposition of Prime Ideals in Extensions..........129
4.2.4 Decomposition of primes in cyslotomic fields..........131
4.2.5 Prime Ideals, Valuations and Absolute Values.........132
4.3 Local and Global Methods............................... . 134
4.3.1 p-adic Numbers...................................134
4.3.2 Applications of p-adic Numbers to Solving Congruences 138
4.3.3 The Hubert Symbol ...............................139
4.3.4 Algebraic Extensions of Qp, and the Täte Field........142
4.3.5 Normalized Absolute Values........................143
4.3.6 Places of Number Fields and the Product Formula.....145
4.3.7 Adeles and Ideles..................................146
The Ring of Adeles................................146
The Idele Group ..................................149
4.3.8 The Geometry of Adeles and Ideles..................149
4.4 Class Field Theory ......................................155
4.4.1 Abelian Extensions of the Field of Rational Numbers . . 155
4.4.2 Frobenius Automorphisms of Number Fields and
Artin s Reciprocity Map............................157
4.4.3 The Chebotarev Density Theorem...................159
4.4.4 The Decomposition Law and
the Artin Reciprocity Map .........................159
4.4.5 The Kernel of the Reciprocity Map..................160
4.4.6 The Artin Symbol.................................161
4.4.7 Global Properties of the Artin Symbol...............162
4.4.8 A Link Between the Artin Symbol and Local Symbols. . 163
4.4.9 Properties of the Local Symbol......................164
4.4.10 An Explicit Construction of Abelian Extensions of a
Local Field, and a Calculation of the Local Symbol .... 165
4.4.11 Abelian Extensions of Number Fields................168
4.5 Galois Group in Arithetical Problems......................172
4.5.1 Dividing a circle into n equal parts..................172
4.5.2 Kummer Extensions and the Power Residue Symbol .. . 175
4.5.3 Galois Cohomology................................178
4.5.4 A Cohomological Definition of the Local Symbol......182
4.5.5 The Brauer Group, the Reciprocity Law and the
Minkowski-Hasse Principle.........................184
Contents XI
Arithmetic of algebraic varieties...........................191
5.1 Arithmetic Varieties and Basic Notions of Algebraic Geometry 191
5.1.1 Equations and Rings...............................191
5.1.2 The set of Solutions of a System.....................191
5.1.3 Example: The Language of Congruences..............192
5.1.4 Equivalence of Systems of Equations.................192
5.1.5 Solutions as iC-algebra Homomorphisms..............192
5.1.6 The Spectrum of A Ring...........................193
5.1.7 Regulär Functions.................................193
5.1.8 A Topology on Spec(vl)............................193
5.1.9 Schemes..........................................196
5.1.10 Ring-Valued Points of Schemes......................197
5.1.11 Solutions to Equations and Points of Schemes.........198
5.1.12 Chevalley s Theorem...............................199
5.1.13 Some Geometrie Notions...........................199
5.2 Geometrie Notions in the Study of Diophantine equations .... 202
5.2.1 Basic Questions...................................202
5.2.2 Geometrie classification............................203
5.2.3 Existence of Rational Points and Obstructions to the
Hasse Principle ...................................204
5.2.4 Finite and Infinite Sets of Solutions..................206
5.2.5 Number of points of bounded height.................208
5.2.6 Height and Arakelov Geometry......................211
5.3 Elliptic curves, Abelian Varieties, and Linear Groups.........213
5.3.1 Algebraic Curves and Riemann Surfaces..............213
5.3.2 Elliptic Curves....................................213
5.3.3 Täte Curve and Its Points of Finite Order............219
5.3.4 The Mordeil - Weil Theorem and Galois Cohomology .. 221
5.3.5 Abelian Varieties and Jacobians.....................226
5.3.6 The Jacobian of an Algebraic Curve.................228
5.3.7 Siegel s Formula and Tamagawa Measure.............231
5.4 Diophantine Equations and Galois Representations..........238
5.4.1 The Täte Module of an Elliptic Curve................238
5.4.2 The Theory of Complex Multiplication...............240
5.4.3 Characters of Z-adic Representations.................242
5.4.4 Representations in Positive Characteristic............243
5.4.5 The Täte Module of a Number Field.................244
5.5 The Theorem of Faltings and Finiteness Problems in
Diophantine Geometry...................................247
5.5.1 Reduction of the Mordeil Conjecture to the finiteness
Conjecture .......................................247
5.5.2 The Theorem of Shafarevich on Finiteness for Elliptic
Curves...........................................249
5.5.3 Passage to Abelian varieties ........................250
5.5.4 Finiteness Problems and Tate s Conjecture...........252
XII Contents
5.5.5 Reduction of the conjectures of Täte to the finiteness
properties for isogenies.............................253
5.5.6 The Faltings-Arakelov Height.......................255
5.5.7 Heights under isogenies and Conjecture T............257
6 Zeta Functions and Modular Forms........................261
6.1 Zeta Functions of Arithmetic Scheines......................261
6.1.1 Zeta Functions of Arithmetic Scheines ...............261
6.1.2 Analytic Continuation of the Zeta Functions..........263
6.1.3 Schemes over Finite Fields and Deligne s Theorem.....263
6.1.4 Zeta Functions and Exponential Sums ...............267
6.2 L-Functions, the Theory of Täte and Explicit Formulae ......272
6.2.1 L-Functions of Rational Galois Representations.......272
6.2.2 The Formalism of Artin............................274
6.2.3 Example: The Dedekind Zeta Function...............276
6.2.4 Hecke Characters and the Theory of Täte ............278
6.2.5 Explicit Formulae.................................285
6.2.6 The Weil Group and its Representations .............288
6.2.7 Zeta Functions, L-Functions and Motives.............290
6.3 Modular Forms and Euler Products........................296
6.3.1 A Link Between Algebraic Varieties and L-Functions.. 296
6.3.2 Classical modular forms............................296
6.3.3 Application: Täte Curve and Semistable Elliptic Curves 299
6.3.4 Analytic families of elliptic curves and congruence
subgroups........................................301
6.3.5 Modular forms for congruence subgroups.............302
6.3.6 Hecke Theory.....................................304
6.3.7 Primitive Forms...................................310
6.3.8 Weil s Inverse Theorem ............................312
6.4 Modular Forms and Galois Representations.................317
6.4.1 Ramanujan s congruence and Galois Representations. . . 317
6.4.2 A Link with Eichler-Shimura s Construction..........319
6.4.3 The Shimura^Taniyama-Weil Conjecture.............320
6.4.4 The Conjecture of Birch and Swinnerton-Dyer........321
6.4.5 The Artin Conjecture and Cusp Forms...............327
The Artin conductor...............................329
6.4.6 Modular Representations over Finite Fields...........330
6.5 Automorphic Forms and The Langlands Program............332
6.5.1 A Relation Between Classical Modular Forms and
Representation Theory.............................332
6.5.2 Automorphic L-Functions..........................335
Further analytic properties of automorphic i-functions . 338
6.5.3 The Langlands Functoriality Principle ...............338
6.5.4 Automorphic Forms and Langlands Conjectures.......339
Contents XIII
Fermat s Last Theorem and Families of Modular Forms. ... 341
7.1 Shimura-Taniyama-Weil Conjecture and Reciprocity Laws . . . 341
7.1.1 Problem of Pierre de Fermat (1601-1665).............341
7.1.2 G.Lame s Mistake.................................342
7.1.3 A short overview of Wiles Marvelous Proof...........343
7.1.4 The STW Conjecture..............................344
7.1.5 A connection with the Quadratic Reciprocity Law.....345
7.1.6 A complete proof of the STW conjecture.............345
7.1.7 Modularity of semistable elliptic curves ..............348
7.1.8 Structure of the proof of theorem 7.13 (Semistable
STW Conjecture) .................................349
7.2 Theorem of Langlands-Tunneil and
Modularity Modulo 3....................................352
7.2.1 Galois representations: preparation..................352
7.2.2 Modularity modulo p..............................354
7.2.3 Passage from cusp forms of weight one to cusp forms
of weight two.....................................355
7.2.4 Preliminary review of the stages of the proof of
Theorem 7.13 on modularity........................356
7.3 Modularity of Galois representations and Universal
Deformation Rings ......................................357
7.3.1 Galois Representations over local Noetherian algebras . . 357
7.3.2 Deformations of Galois Representations..............357
7.3.3 Modular Galois representations .....................359
7.3.4 Admissible Deformations and Modular Deformations. .. 361
7.3.5 Universal Deformation Rings........................363
7.4 Wiles Main Theorem and Isomorphism Criteria for Local
Rings..................................................365
7.4.1 Strategy of the proof of the Main Theorem 7.33.......365
7.4.2 Surjectivity of ps.................................365
7.4.3 Constructions of the universal deformation ring Rs .... 367
7.4.4 A sketch of a construction of the universal modular
deformation ring Tz...............................368
7.4.5 Universality and the Chebotarev density theorem......369
7.4.6 Isomorphism Criteria for local rings..................370
7.4.7 J-structures and the second criterion of isomorphism
of local rings......................................371
7.5 Wiles Induction Step: Application of the Criteria and Galois
Cohomology............................................373
7.5.1 Wiles induction step in the proof of
Main Theorem 7.33................................373
7.5.2 A formula relating ##1^ and ##rw : preparation.....374
7.5.3 The Seltner group and #/jE.........................375
7.5.4 Infinitesimal deformations..........................375
7.5.5 Deformations of type VE...........................377
XIV Contents
7.6 The Relative Invariant, the Main Inequality and The Minimal
Case...................................................382
7.6.1 The Relative invariant.............................382
7.6.2 The Main Inequality...............................383
7.6.3 The Minimal Case.................................386
7.7 End of Wiles Proof and Theorem on Absolute Irreducibility . . 388
7.7.1 Theorem on Absolute Irreducibility..................388
7.7.2 From p = 3 to p = 5...............................390
7.7.3 Families of elliptic curves with fixed ~p5 E.............391
7.7.4 The end of the proof...............................392
The most important insights........................393
Part III Analogies and Visions
III-O Introductory survey to part III: motivations and
description ................................................397
III. 1 Analogies and differences between numbers and functions:
oo-point, Archimedean properties etc.......................397
III. 1.1 Cauchy residue formula and the product formula......397
III. 1.2 Arithmetic varieties................................398
III. 1.3 Infinitesimal neighborhoods of fibers.................398
111.2 Arakelov geometry, fiber over oo, cycles, Green functions
(d apres Gillet-Soule) ....................................399
111.2.1 Arithmetic Chow groups ...........................400
111.2.2 Arithmetic intersection theory and arithmetic
Riemann-Roch theorem............................401
111.2.3 Geometrie description of the closed fibers at infinity . . . 402
111.3 C-functions, local factors at oo, Serre s -T-factors.............404
111.3.1 Archimedean L-factors.............................405
111.3.2 Deninger s formulae ...............................406
111.4 A guess that the missing geometric objeets are
noncommutative spaces ..................................407
111.4.1 Types and examples of noncommutative spaces, and
how to work with them. Noncommutative geometry
and arithmetic....................................407
Isomorphism of noncommutative spaces and Morita
equivalence ...............................409
The tools of noncommutative geometry..............410
111.4.2 Generalities on spectral triples......................411
111.4.3 Contents of Part III: description of parts of this program412
Contents XV
8 Arakelov Geometry and Noncommutative Geometry.......415
8.1 Schottky Uniformization and Arakelov Geometry............415
8.1.1 Motivations and the context of the work of
Consani-Marcolli..................................415
8.1.2 Analytic construction of degenerating curves over
complete local fields ...............................416
8.1.3 Schottky groups and new perspectives in Arakelov
geometry.........................................420
Schottky uniformization and Schottky groups.........421
Fuchsian and Schottky uniformization................424
8.1.4 Hyperbolic handlebodies ...........................425
Geodesics in Xp ..................................427
8.1.5 Arakelov geometry and hyperbolic geometry..........427
Arakelov Green function ...........................427
Cross ratio and geodesics...........................428
Differentials and Schottky uniformization.............428
Green function and geodesics.......................430
8.2 Cohomological Constructions .............................431
8.2.1 Archimedean cohomology...........................431
Operators........................................433
SL(2, E) representations............................434
8.2.2 Local factor and Archimedean cohomology ...........435
8.2.3 Cohomological constructions........................436
8.2.4 Zeta function of the Special fiber and Reidemeister
torsion...........................................437
8.3 Spectral Triples, Dynamics and Zeta Functions..............440
8.3.1 A dynamical theory at infinity......................442
8.3.2 Homotopy quotion.................................443
8.3.3 Filtration ........................................444
8.3.4 Hubert space and grading..........................446
8.3.5 Cuntz-Krieger algebra.............................446
Spectral triples for Schottky groups..................448
8.3.6 Arithmetic surfaces: homology and cohomology .......449
8.3.7 Archimedean factors from dynamics.................450
8.3.8 A Dynamical theory for Mumford curves.............451
Genus two example................................452
8.3.9 Cohomology of W{A/r)T..........................454
8.3.10 Spectral triples and Mumford curves.................456
8.4 Reduction mod oo.......................................458
8.4.1 Homotopy quotients and reduction mod infinity .....458
8.4.2 Baum-Connes map................................460
References.....................................................461
Index..........................................................503
|
| adam_txt |
Contents
Part I Problems and Tricks
Elementary Number Theory. 9
1.1 Problems About Primes. Divisibility and Primality. 9
1.1.1 Arithmetical Notation . 9
1.1.2 Primes and composite numbers . 10
1.1.3 The Factorization Theorem and the Euclidean
Algorithm. 12
1.1.4 Calculations with Residue Classes. 13
1.1.5 The Quadratic Reciprocity Law and Its Use . 15
1.1.6 The Distribution of Primes. 17
1.2 Diophantine Equations of Degree One and Two. 22
1.2.1 The Equation ax + by = c. 22
1.2.2 Linear Diophantine Systems. 22
1.2.3 Equations of Degree Two. 24
1.2.4 The Minkowski-Hasse Principle for Quadratic Forms. 26
1.2.5 Pell's Equation. 28
1.2.6 Representation of Integers and Quadratic Forms by
Quadratic Forms. 29
1.2.7 Analytic Methods. 33
1.2.8 Equivalence of Binary Quadratic Forms. 35
1.3 Cubic Diophantine Equations. 38
1.3.1 The Problem of the Existence of a Solution. 38
1.3.2 Addition of Points on a Cubic Curve. 38
1.3.3 The Structure of the Group of Rational Points of a
Non-Singular Cubic Curve . 40
1.3.4 Cubic Congruences Modulo a Prime. 47
1.4 Approximations and Continued Fractions. 50
1.4.1 Best Approximations to Irrational Numbers . 50
1.4.2 Farey Series. 50
1.4.3 Continued Fractions. 51
VIII Contents
1.4.4 SL2-Equivalence. 53
1.4.5 Periodic Continued Fractions and Pell's Equation. 53
1.5 Diophantine Approximation and the Irrationality. 55
1.5.1 Ideas in the Proof that ((3) is Irrational. 55
1.5.2 The Measure of Irrationality of a Number. 56
1.5.3 The Thue-Siegel-Roth Theorem, Transcendental
Numbers, and Diophantine Equations. 57
1.5.4 Proofs of the Identities (1.5.1) and (1.5.2). 58
1.5.5 The Recurrent Sequences an and bn . 59
1.5.6 Transcendental Numbers and the Seventh Hubert
Problem. 61
1.5.7 Work of Yu.V. Nesterenko on e*, [Nes99]. 61
2 Some Applications of Elementary Number Theory. 63
2.1 Factorization and Public Key Cryptosystems. 63
2.1.1 Factorization is Time-Consuming. 63
2.1.2 One-Way Functions and Public Key Encryption. 63
2.1.3 A Public Key Cryptosystem. 64
2.1.4 Statistics and Mass Production of Primes. 66
2.1.5 Probabilistic Primality Tests. 66
2.1.6 The Discrete Logarithm Problem and The
Diffie-Hellman Key Exchange Protocol. 67
2.1.7 Computing of the Discrete Logarithm on Elliptic
Curves over Finite Fields (ECDLP). 68
2.2 Deterministic Primality Tests. 69
2.2.1 Adleman-Pomerance-Rumely Primality Test: Basic
Ideas . 69
2.2.2 Gauss Sums and Their Use in Primality Testing. 71
2.2.3 Detailed Description of the Primality Test. 75
2.2.4 Primes is in P . 78
2.2.5 The algorithm of M. Agrawal, N. Kayal and N. Saxena . 81
2.2.6 Practical and Theoretical Primality Proving. The
ECPP (Elliptic Curve Primality Proving by F.Morain,
see [AtMo93b]). 81
2.2.7 Primes in Arithmetic Progression. 82
2.3 Factorization of Large Integers. 84
2.3.1 Comparative Difficulty of Primality Testing and
Factorization . 84
2.3.2 Factorization and Quadratic Forms. 84
2.3.3 The Probabilistic Algorithm CLASNO. 85
2.3.4 The Continued Fractions Method (CFRAC) and Real
Quadratic Fields. 87
2.3.5 The Use of Elliptic Curves. 90
Contents IX
Part II Ideas and Theories
3 Induction and Recursion. 95
3.1 Elementary Number Theory From the Point of View of Logic . 95
3.1.1 Elementary Number Theory. 95
3.1.2 Logic. 96
3.2 Diophantine Sets . 98
3.2.1 Enumerability and Diophantine Sets . 98
3.2.2 Diophantineness of enumerable sets. 98
3.2.3 First properties of Diophantine sets . 98
3.2.4 Diophantineness and Pell's Equation. 99
3.2.5 The Graph of the Exponent is Diophantine.100
3.2.6 Diophantineness and Binomial coefficients.100
3.2.7 Binomial coefficients as remainders.101
3.2.8 Diophantineness of the Factorial.101
3.2.9 Factorial and Euclidean Division.101
3.2.10 Supplementary Results.102
3.3 Partially Recursive Functions and Enumerable Sets.103
3.3.1 Partial Functions and Computable Functions .103
3.3.2 The Simple Functions.103
3.3.3 Elementary Operations on Partial functions.103
3.3.4 Partially Recursive Description of a Function.104
3.3.5 Other Recursive Functions.106
3.3.6 Further Properties of Recursive Functions.108
3.3.7 Link with Level Sets.108
3.3.8 Link with Projections of Level Sets.108
3.3.9 Matiyasevich's Theorem.109
3.3.10 The existence of certain bijections.109
3.3.11 Operations on primitively enumerable sets.111
3.3.12 Gödel's function.111
3.3.13 Discussion of the Properties of Enumerable Sets.112
3.4 Diophantineness of a Set and algorithmic Undecidability.113
3.4.1 Algorithmic undecidability and unsolvability.113
3.4.2 Sketch Proof of the Matiyasevich Theorem .113
4 Arithmetic of algebraic numbers.115
4.1 Algebraic Numbers: Their Realizations and Geometry.115
4.1.1 Adjoining Roots of Polynomials.115
4.1.2 Galois Extensions and Frobenius Elements.117
4.1.3 Tensor Products of Fields and Geometrie Realizations
of Algebraic Numbers.119
4.1.4 Units, the Logarithmic Map, and the Regulator.121
4.1.5 Lattice Points in a Convex Body.123
Contents
4.1.6 Deduction of Dirichlet's Theorem From Minkowski's
Lemma .125
4.2 Decomposition of Prime Ideals, Dedekind Domains, and
Valuations.126
4.2.1 Prime Ideals and the Unique Factorization Property . . . 126
4.2.2 Finiteness of the Class Number .128
4.2.3 Decomposition of Prime Ideals in Extensions.129
4.2.4 Decomposition of primes in cyslotomic fields.131
4.2.5 Prime Ideals, Valuations and Absolute Values.132
4.3 Local and Global Methods. . 134
4.3.1 p-adic Numbers.134
4.3.2 Applications of p-adic Numbers to Solving Congruences 138
4.3.3 The Hubert Symbol .139
4.3.4 Algebraic Extensions of Qp, and the Täte Field.142
4.3.5 Normalized Absolute Values.143
4.3.6 Places of Number Fields and the Product Formula.145
4.3.7 Adeles and Ideles.146
The Ring of Adeles.146
The Idele Group .149
4.3.8 The Geometry of Adeles and Ideles.149
4.4 Class Field Theory .155
4.4.1 Abelian Extensions of the Field of Rational Numbers . . 155
4.4.2 Frobenius Automorphisms of Number Fields and
Artin's Reciprocity Map.157
4.4.3 The Chebotarev Density Theorem.159
4.4.4 The Decomposition Law and
the Artin Reciprocity Map .159
4.4.5 The Kernel of the Reciprocity Map.160
4.4.6 The Artin Symbol.161
4.4.7 Global Properties of the Artin Symbol.162
4.4.8 A Link Between the Artin Symbol and Local Symbols. . 163
4.4.9 Properties of the Local Symbol.164
4.4.10 An Explicit Construction of Abelian Extensions of a
Local Field, and a Calculation of the Local Symbol . 165
4.4.11 Abelian Extensions of Number Fields.168
4.5 Galois Group in Arithetical Problems.172
4.5.1 Dividing a circle into n equal parts.172
4.5.2 Kummer Extensions and the Power Residue Symbol . . 175
4.5.3 Galois Cohomology.178
4.5.4 A Cohomological Definition of the Local Symbol.182
4.5.5 The Brauer Group, the Reciprocity Law and the
Minkowski-Hasse Principle.184
Contents XI
Arithmetic of algebraic varieties.191
5.1 Arithmetic Varieties and Basic Notions of Algebraic Geometry 191
5.1.1 Equations and Rings.191
5.1.2 The set of Solutions of a System.191
5.1.3 Example: The Language of Congruences.192
5.1.4 Equivalence of Systems of Equations.192
5.1.5 Solutions as iC-algebra Homomorphisms.192
5.1.6 The Spectrum of A Ring.193
5.1.7 Regulär Functions.193
5.1.8 A Topology on Spec(vl).193
5.1.9 Schemes.196
5.1.10 Ring-Valued Points of Schemes.197
5.1.11 Solutions to Equations and Points of Schemes.198
5.1.12 Chevalley's Theorem.199
5.1.13 Some Geometrie Notions.199
5.2 Geometrie Notions in the Study of Diophantine equations . 202
5.2.1 Basic Questions.202
5.2.2 Geometrie classification.203
5.2.3 Existence of Rational Points and Obstructions to the
Hasse Principle .204
5.2.4 Finite and Infinite Sets of Solutions.206
5.2.5 Number of points of bounded height.208
5.2.6 Height and Arakelov Geometry.211
5.3 Elliptic curves, Abelian Varieties, and Linear Groups.213
5.3.1 Algebraic Curves and Riemann Surfaces.213
5.3.2 Elliptic Curves.213
5.3.3 Täte Curve and Its Points of Finite Order.219
5.3.4 The Mordeil - Weil Theorem and Galois Cohomology . 221
5.3.5 Abelian Varieties and Jacobians.226
5.3.6 The Jacobian of an Algebraic Curve.228
5.3.7 Siegel's Formula and Tamagawa Measure.231
5.4 Diophantine Equations and Galois Representations.238
5.4.1 The Täte Module of an Elliptic Curve.238
5.4.2 The Theory of Complex Multiplication.240
5.4.3 Characters of Z-adic Representations.242
5.4.4 Representations in Positive Characteristic.243
5.4.5 The Täte Module of a Number Field.244
5.5 The Theorem of Faltings and Finiteness Problems in
Diophantine Geometry.247
5.5.1 Reduction of the Mordeil Conjecture to the finiteness
Conjecture .247
5.5.2 The Theorem of Shafarevich on Finiteness for Elliptic
Curves.249
5.5.3 Passage to Abelian varieties .250
5.5.4 Finiteness Problems and Tate's Conjecture.252
XII Contents
5.5.5 Reduction of the conjectures of Täte to the finiteness
properties for isogenies.253
5.5.6 The Faltings-Arakelov Height.255
5.5.7 Heights under isogenies and Conjecture T.257
6 Zeta Functions and Modular Forms.261
6.1 Zeta Functions of Arithmetic Scheines.261
6.1.1 Zeta Functions of Arithmetic Scheines .261
6.1.2 Analytic Continuation of the Zeta Functions.263
6.1.3 Schemes over Finite Fields and Deligne's Theorem.263
6.1.4 Zeta Functions and Exponential Sums .267
6.2 L-Functions, the Theory of Täte and Explicit Formulae .272
6.2.1 L-Functions of Rational Galois Representations.272
6.2.2 The Formalism of Artin.274
6.2.3 Example: The Dedekind Zeta Function.276
6.2.4 Hecke Characters and the Theory of Täte .278
6.2.5 Explicit Formulae.285
6.2.6 The Weil Group and its Representations .288
6.2.7 Zeta Functions, L-Functions and Motives.290
6.3 Modular Forms and Euler Products.296
6.3.1 A Link Between Algebraic Varieties and L-Functions. 296
6.3.2 Classical modular forms.296
6.3.3 Application: Täte Curve and Semistable Elliptic Curves 299
6.3.4 Analytic families of elliptic curves and congruence
subgroups.301
6.3.5 Modular forms for congruence subgroups.302
6.3.6 Hecke Theory.304
6.3.7 Primitive Forms.310
6.3.8 Weil's Inverse Theorem .312
6.4 Modular Forms and Galois Representations.317
6.4.1 Ramanujan's congruence and Galois Representations. . . 317
6.4.2 A Link with Eichler-Shimura's Construction.319
6.4.3 The Shimura^Taniyama-Weil Conjecture.320
6.4.4 The Conjecture of Birch and Swinnerton-Dyer.321
6.4.5 The Artin Conjecture and Cusp Forms.327
The Artin conductor.329
6.4.6 Modular Representations over Finite Fields.330
6.5 Automorphic Forms and The Langlands Program.332
6.5.1 A Relation Between Classical Modular Forms and
Representation Theory.332
6.5.2 Automorphic L-Functions.335
Further analytic properties of automorphic i-functions . 338
6.5.3 The Langlands Functoriality Principle .338
6.5.4 Automorphic Forms and Langlands Conjectures.339
Contents XIII
Fermat's Last Theorem and Families of Modular Forms. . 341
7.1 Shimura-Taniyama-Weil Conjecture and Reciprocity Laws . . . 341
7.1.1 Problem of Pierre de Fermat (1601-1665).341
7.1.2 G.Lame's Mistake.342
7.1.3 A short overview of Wiles' Marvelous Proof.343
7.1.4 The STW Conjecture.344
7.1.5 A connection with the Quadratic Reciprocity Law.345
7.1.6 A complete proof of the STW conjecture.345
7.1.7 Modularity of semistable elliptic curves .348
7.1.8 Structure of the proof of theorem 7.13 (Semistable
STW Conjecture) .349
7.2 Theorem of Langlands-Tunneil and
Modularity Modulo 3.352
7.2.1 Galois representations: preparation.352
7.2.2 Modularity modulo p.354
7.2.3 Passage from cusp forms of weight one to cusp forms
of weight two.355
7.2.4 Preliminary review of the stages of the proof of
Theorem 7.13 on modularity.356
7.3 Modularity of Galois representations and Universal
Deformation Rings .357
7.3.1 Galois Representations over local Noetherian algebras . . 357
7.3.2 Deformations of Galois Representations.357
7.3.3 Modular Galois representations .359
7.3.4 Admissible Deformations and Modular Deformations. . 361
7.3.5 Universal Deformation Rings.363
7.4 Wiles' Main Theorem and Isomorphism Criteria for Local
Rings.365
7.4.1 Strategy of the proof of the Main Theorem 7.33.365
7.4.2 Surjectivity of ps.365
7.4.3 Constructions of the universal deformation ring Rs . 367
7.4.4 A sketch of a construction of the universal modular
deformation ring Tz.368
7.4.5 Universality and the Chebotarev density theorem.369
7.4.6 Isomorphism Criteria for local rings.370
7.4.7 J-structures and the second criterion of isomorphism
of local rings.371
7.5 Wiles' Induction Step: Application of the Criteria and Galois
Cohomology.373
7.5.1 Wiles' induction step in the proof of
Main Theorem 7.33.373
7.5.2 A formula relating ##1^ and ##rw : preparation.374
7.5.3 The Seltner group and #/jE.375
7.5.4 Infinitesimal deformations.375
7.5.5 Deformations of type VE.377
XIV Contents
7.6 The Relative Invariant, the Main Inequality and The Minimal
Case.382
7.6.1 The Relative invariant.382
7.6.2 The Main Inequality.383
7.6.3 The Minimal Case.386
7.7 End of Wiles' Proof and Theorem on Absolute Irreducibility . . 388
7.7.1 Theorem on Absolute Irreducibility.388
7.7.2 From p = 3 to p = 5.390
7.7.3 Families of elliptic curves with fixed ~p5 E.391
7.7.4 The end of the proof.392
The most important insights.393
Part III Analogies and Visions
III-O Introductory survey to part III: motivations and
description .397
III. 1 Analogies and differences between numbers and functions:
oo-point, Archimedean properties etc.397
III. 1.1 Cauchy residue formula and the product formula.397
III. 1.2 Arithmetic varieties.398
III. 1.3 Infinitesimal neighborhoods of fibers.398
111.2 Arakelov geometry, fiber over oo, cycles, Green functions
(d'apres Gillet-Soule) .399
111.2.1 Arithmetic Chow groups .400
111.2.2 Arithmetic intersection theory and arithmetic
Riemann-Roch theorem.401
111.2.3 Geometrie description of the closed fibers at infinity . . . 402
111.3 C-functions, local factors at oo, Serre's -T-factors.404
111.3.1 Archimedean L-factors.405
111.3.2 Deninger's formulae .406
111.4 A guess that the missing geometric objeets are
noncommutative spaces .407
111.4.1 Types and examples of noncommutative spaces, and
how to work with them. Noncommutative geometry
and arithmetic.407
Isomorphism of noncommutative spaces and Morita
equivalence .409
The tools of noncommutative geometry.410
111.4.2 Generalities on spectral triples.411
111.4.3 Contents of Part III: description of parts of this program412
Contents XV
8 Arakelov Geometry and Noncommutative Geometry.415
8.1 Schottky Uniformization and Arakelov Geometry.415
8.1.1 Motivations and the context of the work of
Consani-Marcolli.415
8.1.2 Analytic construction of degenerating curves over
complete local fields .416
8.1.3 Schottky groups and new perspectives in Arakelov
geometry.420
Schottky uniformization and Schottky groups.421
Fuchsian and Schottky uniformization.424
8.1.4 Hyperbolic handlebodies .425
Geodesics in Xp .427
8.1.5 Arakelov geometry and hyperbolic geometry.427
Arakelov Green function .427
Cross ratio and geodesics.428
Differentials and Schottky uniformization.428
Green function and geodesics.430
8.2 Cohomological Constructions .431
8.2.1 Archimedean cohomology.431
Operators.433
SL(2, E) representations.434
8.2.2 Local factor and Archimedean cohomology .435
8.2.3 Cohomological constructions.436
8.2.4 Zeta function of the Special fiber and Reidemeister
torsion.437
8.3 Spectral Triples, Dynamics and Zeta Functions.440
8.3.1 A dynamical theory at infinity.442
8.3.2 Homotopy quotion.443
8.3.3 Filtration .444
8.3.4 Hubert space and grading.446
8.3.5 Cuntz-Krieger algebra.446
Spectral triples for Schottky groups.448
8.3.6 Arithmetic surfaces: homology and cohomology .449
8.3.7 Archimedean factors from dynamics.450
8.3.8 A Dynamical theory for Mumford curves.451
Genus two example.452
8.3.9 Cohomology of W{A/r)T.454
8.3.10 Spectral triples and Mumford curves.456
8.4 Reduction mod oo.458
8.4.1 Homotopy quotients and "reduction mod infinity" .458
8.4.2 Baum-Connes map.460
References.461
Index.503 |
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| author | Manin, Jurij I. 1937-2023 Pančiškin, Aleksej A. 1953- |
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| illustrated | Illustrated |
| index_date | 2024-07-02T19:00:31Z |
| indexdate | 2024-07-09T21:08:20Z |
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| language | English |
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| spelling | Manin, Jurij I. 1937-2023 (DE-588)121177580 aut Introduction to modern number theory fundamental problems, ideas and theories Yuri Ivanovic Manin ; Alexei A. Panchishkin 2. ed., 2. corr. print. Berlin [u.a.] Springer 2007 XV, 514 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopaedia of mathematical sciences 49 Encyclopaedia of mathematical sciences : Number theory 1 Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Pančiškin, Aleksej A. 1953- (DE-588)128761407 aut Encyclopaedia of mathematical sciences 49 (DE-604)BV024126459 49 Encyclopaedia of mathematical sciences Number theory ; 1 (DE-604)BV020831274 1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016152126&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Manin, Jurij I. 1937-2023 Pančiškin, Aleksej A. 1953- Introduction to modern number theory fundamental problems, ideas and theories Encyclopaedia of mathematical sciences Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Introduction to modern number theory fundamental problems, ideas and theories |
| title_auth | Introduction to modern number theory fundamental problems, ideas and theories |
| title_exact_search | Introduction to modern number theory fundamental problems, ideas and theories |
| title_exact_search_txtP | Introduction to modern number theory fundamental problems, ideas and theories |
| title_full | Introduction to modern number theory fundamental problems, ideas and theories Yuri Ivanovic Manin ; Alexei A. Panchishkin |
| title_fullStr | Introduction to modern number theory fundamental problems, ideas and theories Yuri Ivanovic Manin ; Alexei A. Panchishkin |
| title_full_unstemmed | Introduction to modern number theory fundamental problems, ideas and theories Yuri Ivanovic Manin ; Alexei A. Panchishkin |
| title_short | Introduction to modern number theory |
| title_sort | introduction to modern number theory fundamental problems ideas and theories |
| title_sub | fundamental problems, ideas and theories |
| topic | Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie |
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