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Silverman J. — The arithmetic of dynamical systems
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.



Язык: en

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
$GL_{2}$      10
$H^{1}$      202 236
$j$-invariant      221 337
$j$-invariant, in terms of cross-ratio      348
$l$-adic representation      344
$m$-torsion subgroup of elliptic curve      343
$n$-multiplier spectrum      182
$n$-multiplier spectrum, formal      182
$n$-multiplier spectrum, of $z^{d}$      182
$n$-period polynomial      149 181
$n^{th}$-dynatomic polynomial      149 181
$n^{th}$-dynatomic polynomial, homogeneous of degree $\nu_{d}{n)$      181
$O(1)$      93
$ord_{p}$      82
$ord_{p}$, of a polynomial      218
$ord_{\alpha}$      12
$p$-adic absolute value      44
$p$-adic Cauchy residue formula      314
$p$-adic hyperbolic map      279 285 317
$p$-adic inverse function theorem      312
$Per_{n}(0)$      18
$PGL_{2}$      10
$PGL_{2}$, $PGL_{2}(K)$ equivalence      195 233
$PGL_{2}$, action of Galois      203
$PGL_{2}$, automorphism group is subgroup      234
$PGL_{2}$, cocycle gives twist of $P^{1}$      211 215
$PGL_{2}$, conjugation is algebraic group action      173
$PGL_{2}$, equivalence      195
$PGL_{2}$, extension of cocycle to $Aut(\phi)$      203
$PGL_{2}$, finite subgroups      197
$PGL_{2}$, invariant functions on $Rat_{d}$      175
$PGL_{2}$, is $Rat_{1}$      170
$PGL_{2}$, is group variety      170
$PGL_{2}$, map from $PSL_{2}$      175 231
$PGL_{2}$, minimal resultant is invariant      221
$PGL_{2}$, quotient of $Rat_{d}$ by      174
$PGL_{n}$      226
$PGL_{n}$, map from $PSL_{n}$      231
$Rat^{ss}_{d}$      178 179
$Rat^{s}_{d}$      178 179
$Rat_{1}$ is $PGL_{2}$      170
$Rat_{2}$      170
$Rat_{2}$, normal forms lemma      190 233
$Rat_{2}$, subvariety with $Aut =\mu_{2}$      235
$Rat_{d}$      169
$Rat_{d}$, $PGL_{2}$-invariant functions on      175
$Rat_{d}$, affine coordinate ring      169 174
$Rat_{d}$, coordinate ring contains $\sigma^{(n)}_{i}$ and $\sigma^{*(n)}_{i}$      183
$Rat_{d}$, generic map not very highly ramified      231
$Rat_{d}$, geometry of boundary      170
$Rat_{d}$, in $PSL_{2}$-stable locus      176
$Rat_{d}$, is a rational variety      232
$Rat_{d}$, is subset of $\mathbb{P}^{2d+1}$      169
$Rat_{d}$, map induced by Schwarzian derivative      232
$Rat_{d}$, map to $\mathcal{M}_{d}$      174
$Rat_{d}$, multiplier 1 is Zariski closed subset      230
$Rat_{d}$, quotient by $PGL_{2}$      174
$Rat_{d}$, quotient by $PSL_{2}$ is a variety      175
$Rat_{d}$, semistable rational maps      178
$Rat_{d}$, stable rational maps      178
$Rat_{d}$, subvariety with nontrivial automorphisms      199
$Rat_{d}$, universal map over      171
$S$-unit in finite extension      132
$SL_{2}$      175
$V -E + F$ formula      13 37
$v$-adic chordal metric      45 144
$v$-adic chordal metric, effect of linear fractional transformation      76
$v$-adic chordal metric, invariant maps for      46
$v$-adic chordal metric, is a metric      45
$v$-adic chordal metric, periodic points      69
$v$-adic chordal metric, relation to reduction      49
$v$-adic chordal metric, resultant measures expansion      56
$X_{0}(4)$, $X_{0}(5)$, $X_{0}(6)$      230
$X_{0}(n)$ (dynatomic curve)      161
$X_{0}(n)$ (dynatomic curve), is irreducible      164
$X_{1}(1)$, $X_{1}(2)$, $X_{1}(3)$ are rational      157
$X_{1}(4)$, $X_{1}(5)$, $X_{1}(6)$      230
$X_{1}(n)$ (dynatomic curve)      157
$X_{1}(n)$ (dynatomic curve), action of $z^{2} + c$ on      161
$X_{1}(n)$ (dynatomic curve), genus      164
$X_{1}(n)$ (dynatomic curve), is irreducible      164
$X_{1}(n)$ (dynatomic curve), is wreath product cover of $\mathbb{P}^{1}$      164
$Y_{0}(n)$ (dynatomic curve)      161
$Y_{1}(n)$ (dynatomic curve)      157
$Y_{1}(n)$ (dynatomic curve), action of $z^{2} + c$ on      161
$Y_{1}(n)$ (dynatomic curve), is nonsingular      164
$z^{2} + c$      20
$z^{2} + c$, acts on $X_{1}(n)$      161
$z^{2} + c$, algebraic family      159
$z^{2} + c$, algebraic family of      230
$z^{2} + c$, algebraic family, induces map to dynatomic curve      159
$z^{2} + c$, bifurcation point      165
$z^{2} + c$, bifurcation polynomial      165
$z^{2} + c$, condition for unequal formal and primitive periods      165
$z^{2} + c$, dynatomic curve      157 161
$z^{2} + c$, dynatomic curve, $X_{1}(1)$, $X_{1}(1)$, $X_{1}(1)$ are rational      157
$z^{2} + c$, dynatomic curve, genus of      164
$z^{2} + c$, dynatomic curve, is irreducible      164
$z^{2} + c$, dynatomic curve, is modular curve      158
$z^{2} + c$, dynatomic curve, is nonsingular      164
$z^{2} + c$, dynatomic curve, is wreath product cover of $\mathbb{P}^{1}$      164
$z^{2} + c$, dynatomic polynomial      156
$z^{2} + c$, Mandelbrot set      26 165
$z^{2} + c$, Mandelbrot set, in disk of radius 2      166
$z^{2} + c$, map $X_{1}(2)\rightarrow X_{0}(2)$      162
$z^{2} + c$, map $X_{1}(3)\rightarrow X_{0}(3)$      230
$z^{2} + c$, Misiurewicz point      166
$z^{2} + c$, no neutral cycles      166
$z^{2} + c$, period polynomial      156
$z^{2} + c$, points on dynatomic curve solve moduli problem      159
$z^{2} + c$, quadratic polynomial conjugate to      156
$z^{2} + c$, repelling critical orbit      166
$z^{2} + c$, subhyperbolic      166
$z^{2} + c$, with preperiodic critical point      166
$\acute{e}$tale map      353
$\mathbb{C}_{p}$      239
$\mathbb{C}_{p}$, bounded seminorm on on $\mathbb{C}_{p}[z]$      296
$\mathbb{C}_{p}$, Cauchy sequence      296
$\mathbb{C}_{p}$, compact Julia set example      275
$\mathbb{C}_{p}$, is algebraically closed      239 242
$\mathbb{C}_{p}$, is complete      242
$\mathbb{C}_{p}$, is contained in Berkovich affine line      302
$\mathbb{C}_{p}$, is totally disconnected      239 276 294
$\mathbb{C}_{p}$, not locally compact      239 294
$\mathbb{C}_{p}$, projective space is bounded      243
$\mathbb{C}_{p}$, projective space not compact      243
$\mathbb{C}_{p}$, unit disk contained in Berkovich disk      295 301 323
$\mathbb{C}_{p}$, valuation      249
$\mathbb{C}_{p}$, value group is $\mathbb{Q}$      294
$\mathcal{M}_{2}$      188
$\mathcal{M}_{2}$, completion equals $\mathbb{P}^{2}$      194
$\mathcal{M}_{2}$, explicit formula for $\sigma_{1}$ , $\sigma_{1}$      189
$\mathcal{M}_{2}$, is a rational variety      232
$\mathcal{M}_{2}$, is a scheme over $\mathbb{Z}$      189
$\mathcal{M}_{2}$, isomorphic to $\mathbb{A}^{2}$      188
$\mathcal{M}_{2}$, locus of polynomial maps      232
$\mathcal{M}_{2}$, point with $Aut(\phi) = S_{3}$      234
$\mathcal{M}_{2}$, subvariety with $Aut = \mu_{0}$      235
$\mathcal{M}_{d}$      174
$\mathcal{M}_{d}$, $\mathcal{M}_{2} \cong \mathbb{A}^{2}$      189
$\mathcal{M}_{d}$, $\mathcal{M}_{3}$, embedded by $\sigma^{(n)}_{i}$      232
$\mathcal{M}_{d}$, affine coordinate ring      176
$\mathcal{M}_{d}$, coordinate ring contains $\sigma_{i}^{(n)}$ and $\sigma_{i}^{*(n)}$      183
$\mathcal{M}_{d}$, field of moduli      207
$\mathcal{M}_{d}$, generic map not very highly ramified      231
$\mathcal{M}_{d}$, has dimension $2d - 2$      176
$\mathcal{M}_{d}$, height      221
$\mathcal{M}_{d}$, is a rational variety?      232
$\mathcal{M}_{d}$, is a scheme over $\mathbb{Z}$      179 186
$\mathcal{M}_{d}$, is a unirational variety      232
$\mathcal{M}_{d}$, is a variety      175
$\mathcal{M}_{d}$, is affine      176
$\mathcal{M}_{d}$, is complex orbifold      176
$\mathcal{M}_{d}$, is connected      176
$\mathcal{M}_{d}$, is integral      176
$\mathcal{M}_{d}$, is nonsingular?      193
$\mathcal{M}_{d}$, map from $Rat_{d}$      174 175
$\mathcal{M}_{d}$, multiplier 1 is Zariski closed subset      230
$\mathcal{M}_{d}$, not embedded by $\sigma^{(n)}_{i}$      186 187 356 367
$\mathcal{M}_{d}$, rational points      177
$\mathcal{M}_{d}$, semistable completion      178
$\mathcal{M}_{d}$, stable completion      178
$\mathcal{M}_{d}$, subvariety with nontrivial automorphisms      199 234
$\mathcal{M}_{d}$, twists have same image      198
$\mathcal{M}_{d}^{ss}$      178
$\mathcal{M}_{d}^{ss}$, is a scheme over $\mathbb{Z}$      179
$\mathcal{M}_{d}^{ss}$, is categorical quotient      179
$\mathcal{M}_{d}^{ss}$, is projective      178
$\mathcal{M}_{d}^{ss}$, isomorphic to $\mathcal{M}_{d}^{s}$ iff $d$ is even      178
$\mathcal{M}_{d}^{s}$      178
$\mathcal{M}_{d}^{s}$, has natural quotient property      179
$\mathcal{M}_{d}^{s}$, is a scheme over $\mathbb{Z}$      179
$\mathcal{M}_{d}^{s}$, is geometric quotient      179
$\mathcal{M}_{d}^{s}$, is quasiprojective      178
$\mathcal{M}_{d}^{s}$, isomorphic to $\mathcal{M}^{ss}_{d}$ iff $d$ is even      178
$\sigma^{(n)}_{i}$, isomorphism $\mathcal{M}_{2}\cong\mathbb{A}^{2}$      188
$\sigma^{(n)}_{i}$, of $z^{2} + bz$      183
$\sigma^{(n)}_{i}$, of $z^{d}$      183
$\sigma^{*(n)}_{i}$      183
$\sigma^{*(n)}_{i}$, for Latt$\grave{e}$s map      186
$\sigma^{*(n)}_{i}$, in $\mathbb{Q}[\sigma_{1}, \sigma]$ for $Rat_{2}$      189
$\sigma^{*(n)}_{i}$, is in $\mathbb{Q}[\mathcal{M}_{d}]$      183 232
$\sigma^{*(n)}_{i}$, of $z^{2} + bz$      183
$\sigma^{*(n)}_{i}$, of $z^{d}$      183
$\sigma_{d,N}$      187 232
$\sigma_{d,N}$, degree unknown for $d = 3$      188
$\sigma_{d,N}$, embeds $\mathcal{M}_{2}$      188
$\sigma_{i}(\phi)$      180
$\sigma_{i}(\phi)$, explicit formula for $\sigma_{1}(\phi)$      181
$\sigma_{i}(\phi)$, is in $\mathbb{Q}[\mathcal{M}_{d}]$      180
$\sigma_{i}^{(n)}$      183
$\sigma_{i}^{(n)}$, explicit formula for $\sigma_{1}$, $\sigma_{2}$      189
$\sigma_{i}^{(n)}$, for Latt$\grave{e}$s map      186
$\sigma_{i}^{(n)}$, in $\mathbb{Q}[\sigma_{1}, \sigma_{2}]$ for $Rat_{2}$      189
$\sigma_{i}^{(n)}$, integral over $\mathbb{Q}[Rat_{d}]$      185
$\sigma_{i}^{(n)}$, is in $\mathbb{Q}[\mathcal{M}_{d}]$      183
$\wp$, Weierstrass function      34 345
abc conjecture      372 373
Abelian group, preperiodic point equals torsion point      2 41 326
Abelian group, torsion subgroup      2
Abelian variety      409 442
Abelian variety, Bogomolov conjecture      129
Abelian variety, field of definition      217
Abelian variety, field of moduli      217
Abelian variety, Manin — Mumford conjecture      127
Abelian variety, N$\acute{e}$ron local height      104
Abelian variety, Raynaud’s theorem      127
Absolute height      85
Absolute height, logarithmic      93
Absolute value      43
Absolute value, $p$-adic      44 82
Absolute value, Archimedean      82 83
Absolute value, completion of a field at an      83
Absolute value, effect of polynomial map on      288
Absolute value, equivalent      44
Absolute value, extension formula      83 84
Absolute value, local degree      83
Absolute value, nonarchimedean      44 82 83 242
Absolute value, of a point      90 288
Absolute value, of a polynomial      91 288
Absolute value, on $\mathbb{Q}$      82 312
Absolute value, on function fields      44
Absolute value, product formula      83
Absolute value, standard set of ($M_{K}$)      83
Abstract dynamical system      1
Abstract dynamical system, realizable sequence      6
Addition on an elliptic curve      31
Additive group      30
Additivity of height      408 421
Affine automorphism      390 427
Affine automorphism, algebraically reversible      429
Affine automorphism, algebraically stable      396 433
Affine automorphism, canonical height      431
Affine automorphism, composition of elementary maps and H$\acute{e}$non maps      402
Affine automorphism, composition of involutions      429
Affine automorphism, degree      394
Affine automorphism, degree of inverse      394 395
Affine automorphism, disjoint indeterminacy locus      394
Affine automorphism, dynamical degree      433
Affine automorphism, fixed point      427
Affine automorphism, H$\acute{e}$non map      390
Affine automorphism, height inequality      399 431
Affine automorphism, height of periodic points      402
Affine automorphism, indeterminacy locus      389 392
Affine automorphism, indeterminacy locus, dimension of      394
Affine automorphism, indeterminacy locus, disjoint      394
Affine automorphism, indeterminacy locus, of inverse      392
Affine automorphism, indeterminacy locus, of iterate      394
Affine automorphism, inverse may have different degree      390
Affine automorphism, iterate may have wrong degree      390
Affine automorphism, number of orbits over a finite field      430
Affine automorphism, number of orbits over a finite field periodic points of      394 400
Affine automorphism, regular      391 394 427
Affine automorphism, regular periodic points of      428
Affine automorphism, reversible H$\acute{e}$non map      430
Affine coordinate ring, of $Rat_{d}$      169 174
Affine coordinate ring, of $\mathcal{M}_{2}$      189
Affine coordinate ring, of $\mathcal{M}_{d}$      176 179
Affine coordinate ring, of $\mathcal{M}_{d}$, as $\mathbb{Z}$-scheme      186 189
Affine coordinate ring, of $\mathcal{M}_{d}$, contains $\sigma^{(n)}_{i}$      183
Affine coordinate ring, of $\mathcal{M}_{d}$, contains $\sigma^{*(n)}_{i}$      183 232
Affine line, Berkovich      301
Affine minimal model of a rational map      112 372 385
Affine minimal rational map      372
Affine morphism      375 388
Affine morphism, algebraically stable      396 433
Affine morphism, degree of composition      392
Affine morphism, dynamical degree      397 428
Affine morphism, finite quotient      376
Affine morphism, homogenization      389
Affine morphism, indeterminacy locus      389 392 427
Affine morphism, iterates      376
Affine morphism, jointly regular      397 429
Affine morphism, lift of      389
Affine morphism, uniquely determines endomorphism and translation      376
Affine plane, absolute value      288
Affine plane, lift of rational map      287
Affine plane, projection to projective line      287
Affine plane, with origin removed      287
Affine space, absolute value      90
Affine space, height      397
Affine space, projection to projective space      287
Algebraic closure      85
Algebraic dynamics      6
Algebraic entropy      397
Algebraic family      159 230
Algebraic family, field of definition      159
Algebraic family, induces map to dynatomic curve      159
Algebraic family, of $\mathbb{P}^{1}$’s      171
Algebraic family, of morphisms      171
Algebraic family, of quadratic maps      194
Algebraic geometry      402
Algebraic group      325
Algebraic group, commutative      375
Algebraic group, conjugation is action on      173
Algebraic group, finite quotient      376
Algebraic group, iterates of affine morphism      376
Algebraic group, simple      377
Algebraic group, universal cover      375
Algebraic point in Julia set      40
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