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Dwork B., Gerotto G., Sullivan F.J. — An Introduction to G-Functions
Dwork B., Gerotto G., Sullivan F.J. — An Introduction to G-Functions

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Название: An Introduction to G-Functions

Авторы: Dwork B., Gerotto G., Sullivan F.J.

Аннотация:

Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non- zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s.After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, Andrй, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G- series is again a G-series. This book will be indispensable for those wishing to study the work of Bombieri and Andrй on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.


Язык: en

Рубрика: Математика/Анализ/Специальные функции/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 323

Добавлена в каталог: 09.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Accessory parameter      158
Additive character      56
Additive character, trivial      56
Additive radius of convergence      42
Amice ring      129
Analytic element      128 129
Analytic element, analytic in an annulus      131
Analytic function      117
Analytic function, bounded      117
Annulus of convergence      47
Apparent singularity      171
Artin — Hasse exponential      55
Associated scalar operator      87
Binomial series      49 97 140
Bombieri condition      226
Bombieri condition, local      234
Bombieri estimate      261
Boundary seminorm      117
Bounded polygon      303
Cauchy estimates      114
Cauchy sequence      6
Characteristic of a field      4
Chudnovsky’s Theorem      267
Complete field      6
Completion      7
Cyclic differential module      87
Cyclic vector, Theorem of      89
Denominator      224
Dense      7
Derivation      77
Derivation, non-trivial      77
Dieudonne’s Theorem      54
Differential extension      79
Differential field      77
Differential module      86
Differential module, cyclic      87
Differential module, nilpotent      88
Differential operator      78
Differential operator, globally nilpotent      98
Differential operator, nilpotent      81
Differential operator, trivial      79
Differential system      86
Differential system, nilpotent      88
Differential system, reducible      280
Differential system, trivial      87
Dilogarithm      125
Discrete valuation      21
Dwork — Frobenius Theorem      156
Effective growth, Theorem of      120
Entire function      68
Equivalent norms      9
Equivalent valuations      4
Exponential series      50
Exponents of a system      113
Exponents of an operator      84 98
Extension of a valuation      17
Field      3
Frobenius automorphism      32
Fuchs’ Theorem      101
G-function      xiv
G-operator      xiv
G-operator, irreducible      xiii
G-Series      xiii 264
Galockin condition      227
Gauss norm      10
Gauss’ Lemma      10
Generic disk      93
Generic global inverse radius      226
Generic point      93
Global inverse radius      234 241
Global inverse radius, generic      226
Globally nilpotent operator      98
Hadamard product      202
Height (logarithmic absolute)      223
Height (multiplicative)      284
Hensel’s Lemma      11
Heun’s equation      157
Hypergeometric equation      150
Hypergeometric function      150
Indicial polynomial      84 98
Induced topology      3
Inertial subfield      30
Irregular singularity      101
Katz — Honda Theorem      84
Katz’s Theorem      99
Lame equation      158
Laurent series      46
Line of contact      304
Line of support      28
Liouville number (p-adic)      200
Logarithm      38 102
Matrix solution      87
Maximum Modulus Principle      116
Meromorphic function      68 117
Monodromy map      100
Newton polygon of a Laurent series      47
Newton polygon of a polynomial      24
Newton polygon of a power series      42
Newton’s Lemma      13
Nilpotent, differential module      88
Nilpotent, differential operator      81
Nilpotent, differential system      88 (see also “Globally nilpotent operator”)
Norm      8
Normed space      8
Order of an operator      79
Order, function      24
Ordinary point of a system      170
Ostrowski’s Defect Theorem      21
p-adic, analytic function      117
p-adic, entire function      68
p-adic, integer      7
p-adic, Liouville number      200
p-adic, meromorphic function      68
p-adic, number      7
p-adic, valuation      6
Prepared eigenvalues      106
Prime      222
Prime, complex      223
Prime, finite      222
Prime, infinite      222
Prime, real      223
Product formula      223
Radius of convergence, additive      42
Radius of convergence, of a differential system      94
Radius of convergence, of a Laurent series      254
Radius of convergence, of a power series      41
Ramification index      19
Reducible system      280
Regular singularity of a system      101 113
Regular singularity of an operator      84 98
Relative degree      19
Residue field      5
Rouche’s Theorem      131
Shearing transformation      105 169
Shidlovsky’s Lemma      270
Siegel’s Lemma      284
Singular point of a system      171
Singularity      171
Singularity, apparent      171
Singularity, trivial      171
Size of a local solution      243
Size of a matrix      227
Splitting of an operator      81
Tamely ramified      34
Taylor’s Formula      12
Teichmiiller representative      56
Totally ramified extension      30
Trivial character      56
Trivial differential operator      79
Trivial differential system      87
Trivial singularity      171
Trivial valuation      3
Truncation operator      288
Type of a number      199
Uniform part      108
Unimodular matrix      162
Unramified extension      30
Valuation      3
Valuation group      5
Valuation ideal      5
Valuation ring      5
Valuation, p-adic      6
Valuation, trivial      3
Wildly ramified      34
Wronskian, determinant      77
Wronskian, matrix      77
Zeta function      61
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