Silverman J. — The arithmetic of dynamical systems :: Электронная библиотека попечительского совета мехмата МГУ

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Silverman J. — The arithmetic of dynamical systems

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Название: The arithmetic of dynamical systems

Автор: Silverman J.

Аннотация:

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

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Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 518

Добавлена в каталог: 08.02.2014

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Предметный указатель

Divisor, of degree zero      405
Divisor, on K3 surface      418
Divisor, positive      see Effective divisor
Divisor, principal      320 403 405
Divisor, principal iff degree and sum are zero      339 409
Divisor, pullback      404
Divisor, pullback of linearly equivalent      405
Divisor, pushforward      404
Divisor, rational map associated to      406
Divisor, Riemann - Roch theorem      214
Divisor, support      403
Divisor, very ample      406
Domain of quasiperiodicity      310 311
Domain, no wandering      276 284 285
Douady — Hubbard theorem      167
Double cover      410
Double cover, of elliptic curve      347
Drinfeld module      6
Dual isogeny      340 352
Duplication map, elliptic curve      32 338 351 370
Duplication map, on $\mathbb{Q}/\mathbb{Z}$       168 229
Dynamical Bogomolov conjecture      129
Dynamical degree      397 428 433
Dynamical Galois equidistribution      128
Dynamical Lehmer conjecture      101 138
Dynamical Man in -Mumford conjecture      127
Dynamical system      1
Dynamical system, abstract      1
Dynamical system, algebraic      6
Dynamical system, discrete      1
Dynamical system, integrable      429 430
Dynamical system, iteration of arithmetic functions      6
Dynamical system, on Drinfeld modules      6
Dynamical system, on finite field      2 5
Dynamical system, on finite set      1
Dynamical system, on foliated spaces      6
Dynamical system, on function field      5
Dynamical system, on group      2
Dynamical system, on metric space      3
Dynamical system, on topological group      8
Dynamical system, on topological space      3
Dynamical system, reversible      429
Dynamical system, symbolic      258 314
Dynamical unit      69
Dynamical unit, action of Galois      133
Dynamical unit, for power map      328
Dynamical unit, generic monic polynomial      146
Dynamical unit, generic rational map      146
Dynamical unit, polynomial map      71 132
Dynamical unit, quadratic map      134 145 146
Dynamical unit, rational map      71 72 79 132 133
Dynamically affine morphism      376
Dynatomic curve      156 157 161
Dynatomic curve, $X_{0}(4)$ , $X_{0}(5)$ , $X_{0}(6)$       230
Dynatomic curve, $X_{1}(1)$ , $X_{1}(2)$ ), $X_{1}(3)$ are rational      157
Dynatomic curve, $X_{1}(4)$ , $X_{1}(5)$ , $X_{1}(6)$       230
Dynatomic curve, action of $z^{2} + c$ on      161
Dynatomic curve, genus      164 230
Dynatomic curve, is coarse moduli space      160
Dynatomic curve, is fine moduli space      230
Dynatomic curve, is irreducible      164
Dynatomic curve, is modular curve      158
Dynatomic curve, map $X_{1}(2) \rightarrow X_{0}(2)$       162
Dynatomic curve, map $X_{1}(3) \rightarrow X_{0}(3)$       230
Dynatomic curve, points solve moduli problem      159
Dynatomic curve, quadratic polynomial      156
Dynatomic curve, quotient      161
Dynatomic divisor      150
Dynatomic field      123
Dynatomic field, action of Galois      124
Dynatomic field, Galois group is subgroup of wreath product      125
Dynatomic field, quadratic polynomial      123
Dynatomic field, ramification      129 131
Dynatomic field, unit      129
Dynatomic polynomial      39 148 149 181
Dynatomic polynomial, associated multiplier polynomial      225 227
Dynatomic polynomial, dehomogenized      149
Dynatomic polynomial, discriminant      226 227
Dynatomic polynomial, for $z^{2} + c$       156
Dynatomic polynomial, generalized      227 229
Dynatomic polynomial, homogeneous of degree $\nu_{d}(n)$       181
Dynatomic polynomial, is a polynomial      151 226
Dynatomic polynomial, of       224
Dynatomic polynomial, of $z^{2} + bz$       182
Dynatomic polynomial, order at $P (a^{*}_{P}(n))$       151 225 226
Dynatomic polynomial, quadratic map      39
Dynatomic polynomial, reducible      235
Dynatomic polynomial, resultant of two      165 226 227
Dynatomic zero-cycle      150 226
Dynatomic zero-cycle, is positive      226
Effective divisor      406
Effective divisor, positivity of height      408
Eigenvalue      421
Eigenvalue, involution      438
Elementary symmetric polynomial      87 180
Elimination theory      417
Elliptic curve      30 336 409
Elliptic curve, -invariant      337
Elliptic curve, -series      409
Elliptic curve, -torsion subgroup      343
Elliptic curve, addition law      31
Elliptic curve, automorphism group      340 341
Elliptic curve, automorphism group, explicit description      342
Elliptic curve, canonical height      409
Elliptic curve, complex multiplication      32 341
Elliptic curve, complex uniformization      33 127
Elliptic curve, critical values of quotient      353
Elliptic curve, defined over       337
Elliptic curve, degree map on divisors      339
Elliptic curve, degree of an endomorphism      349
Elliptic curve, degree of an isogeny      340
Elliptic curve, degree two map is even      382
Elliptic curve, determined by critical values of double cover      347
Elliptic curve, discriminant      30 337 383
Elliptic curve, division polynomial      362 383
Elliptic curve, divisor group of      339
Elliptic curve, divisor is principal iff degree and elliptic curve, sum are zero      339 409
Elliptic curve, divisor of a function      339
Elliptic curve, divisor on      338
Elliptic curve, dual isogeny      340 352
Elliptic curve, duplication formula      338
Elliptic curve, duplication map      32 351 370
Elliptic curve, endomorphism ring      31 340 341 378
Elliptic curve, field generated by torsion      344 363
Elliptic curve, flexible Latt $\grave{e}$ s map      355
Elliptic curve, fundamental domain      33 127
Elliptic curve, geometric points      337
Elliptic curve, global minimal Weierstrass equation      343 371
Elliptic curve, good reduction      58 62 342 343 362 383
Elliptic curve, good reduction implies reduction is homomorphism      342
Elliptic curve, group law      337
Elliptic curve, group law, explicit formulas      338
Elliptic curve, group law, is algebraic      338
Elliptic curve, Haar measure      127
Elliptic curve, Hecke correspondence      369
Elliptic curve, height on      409
Elliptic curve, ideal class group maps to CM      350
Elliptic curve, identity element      337
Elliptic curve, integer point      372
Elliptic curve, integral -invariant      370
Elliptic curve, invariant differential      344 359 367
Elliptic curve, inverse of a point      337
Elliptic curve, is a pair $(E, \mathcal{O})$       337
Elliptic curve, is abelian group      338
Elliptic curve, isogeny      339
Elliptic curve, isogeny, is homomorphism      340
Elliptic curve, isogeny, is unramified      340 353
Elliptic curve, isomorphic to Picard group      339
Elliptic curve, isomorphism class      337
Elliptic curve, kernel of endomorphism      343

Elliptic curve, Latt $\grave{e}$ s map      32 97 351
Elliptic curve, Latt $\grave{e}$ s map, preperiodic points      41 352
Elliptic curve, level curve of integrable map      430
Elliptic curve, minimal Weierstrass equation      222 342
Elliptic curve, modified group law      382
Elliptic curve, morphism is isogeny and translation      339
Elliptic curve, multiplication map      31 112 340 383
Elliptic curve, multiplication map, effect on height      409
Elliptic curve, multiplication map, kernel      343
Elliptic curve, noncommutative endomorphism ring      378
Elliptic curve, nonconjugate Latt $\grave{e}$ s maps      354
Elliptic curve, nonsingular      337
Elliptic curve, normalized embedding of endomorphism ring      345
Elliptic curve, point at infinity      336
Elliptic curve, potential good reduction      371
Elliptic curve, principal divisor      339
Elliptic curve, principal divisor, has degree zero      339
Elliptic curve, quasiminimal Weierstrass equation      371 385
Elliptic curve, quotient by automorphism group      346 365
Elliptic curve, rational point      337
Elliptic curve, reduction modulo       342
Elliptic curve, reduction of torsion points      62 343 364
Elliptic curve, subgroup of -rational points      338
Elliptic curve, summation map on divisors      339
Elliptic curve, supersingular      378
Elliptic curve, Szpiro conjecture      221
Elliptic curve, Tate’s algorithm      342
Elliptic curve, torsion equidistributed      127
Elliptic curve, torsion point      41
Elliptic curve, torsion subgroup      32 343 352
Elliptic curve, twist      198 341 374
Elliptic curve, uniform bound for integer points      372
Elliptic curve, uniform boundedness of torsion      97 369
Elliptic curve, universal cover      346
Elliptic curve, Weierstrass $\wp$ function      34 345
Elliptic curve, Weierstrass class      237
Elliptic curve, Weierstrass equation      30 336 409
Elliptic modular curve      163
Endomorphism ring, elliptic curve      340 341 378
Endomorphism ring, kernel      343
Endomorphism ring, multiplicative group      325 378
Endomorphism ring, noncommutative      378
Endomorphism ring, normalized embedding in $\mathbb{C}$       345
Endomorphism ring, order in quadratic imaginary field      341
Endomorphism ring, order in quaternion algebra      341
entropy      6
Entropy, algebraic      397
Equi - Lipschitz      313
Equicontinuity      3 22 254 264 265 271 313
Equicontinuity, Fatou set is maximal open      22
Equicontinuity, is not open condition      40 254
Equicontinuity, on a metric space      22
Equicontinuity, topological      306
Equidistribution      6
Equidistribution, dynamical Galois      128
Equidistribution, Galois of a sequence      128
Equidistribution, of preperiodic points      128
Equidistribution, of small height points      128
Equidistribution, torsion on elliptic curve      127
Equivalence relation from group action      234
Er $\ddot{e}$ menko’s theorem      378
Ergodic theory      6
Essential singularity      23
Euclidean distance versus chordal metric      119
Euler totient function      137 164 182
Euler totient function, iteration of      6
Exact period      1 18 38 150
Exact period, condition for larger formal period      151 165 226
Exact period, rational map has points of different      154 155 196
Exceptional set      16 17 266
Exercise, hard (textbf*)      7
Exercise, unsolved (**)      7
Extension formula      83 84
Fairings’ theorem      107 442
Family of maps      171 194
Fatou domain      240
Fatou set      22 255
Fatou set, all of $\mathbb{P}^{1}$       26 59 239
Fatou set, Berkovich      306 311
Fatou set, canonical height of points in      138
Fatou set, classical contained in Berkovich      306
Fatou set, classification of connected components      28
Fatou set, complement of Julia set      22
Fatou set, component at infinity is completely invariant      40
Fatou set, contains attracting periodic points      40
Fatou set, contains nonrepelling periodic points      256
Fatou set, empty      26 35 361
Fatou set, Herman ring      28
Fatou set, is completely invariant      23 40 266
Fatou set, is nonempty      256 285 314
Fatou set, is open      22
Fatou set, Julia set is boundary      25
Fatou set, local canonical height of points in      141
Fatou set, no wandering disk domains      284 285
Fatou set, no wandering domains      28
Fatou set, nonarchimedean      254
Fatou set, number of connected components      27
Fatou set, of $(z^{2} - z)/p$       283
Fatou set, of $z^{d}$       22
Fatou set, of an iterate      24 255
Fatou set, of commuting maps      378
Fatou set, parabolic component      28
Fatou set, polynomial cannot have Herman ring      28
Fatou set, recurrent critical point is periodic      284 317
Fatou set, set of disk components      283
Fatou set, Siegel disk      28
Fermat’s Little Theorem      2 252 275
Fermat’s sum of two squares theorem      200 236
Field of definition      206
Field of definition, abelian variety      217
Field of definition, cocycle is coboundary      209
Field of definition, contains field of moduli      208
Field of definition, covering map      217
Field of definition, intersection of all is field of moduli      236
Field of definition, not equal to field of moduli      208 213 236
Field of definition, of algebraic family      159
Field of definition, of morphism      89
Field of definition, of point      86 122 145
Field of definition, of quotient variety      161
Field of moduli      177 207
Field of moduli, abelian variety      217
Field of moduli, cocycle      209
Field of moduli, contained in field of definition      208
Field of moduli, covering map      217
Field of moduli, equals field of definition for polynomials      215
Field of moduli, equals field of definition if degree is even      215
Field of moduli, from image in $\mathcal{M}_{d}$       207
Field of moduli, is intersection of all fields of definition      236
Field of moduli, not equal to field of definition      208 213 236
Field of moduli, of a point      145
Field, complete      240
Field, completion      241
Field, extension of complete      241
Field, finite      2
Field, of characteristic       37
Field, separable extension      37
Field, Valued      44 240
Filled Julia set      74 140
Filled Julia set, Green function      140
Fine moduli space      160 172 230
Finite field, dynamical system on      2 5
Finite field, function field over      5
Finite invariant set      16 266
Finite subgroup of $PGL_{2}$       197
Fixed point      18
Fixed point, attracting in Berkovich space      324
Fixed point, flexible Latt $\grave{e}$ s map      382
Fixed point, Galois conjugates      180 184
Fixed point, Galois group      427
Fixed point, index summation formula      20 38 255 314 380 382

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