An Introduction to G-functions

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Princeton University Press, 1994 - Mathematics - 323 pages
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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, André, 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 André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

 

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Contents

Valued Fields 1 Valuations
3
Complete Valued Fields
6
Normed Vector Spaces
8
Hensels Lemma
10
Extensions of Valuations
17
Newton Polygons
24
The intercept Method
28
Ramification Theory
30
Effective Bounds The Dwork Robba Theorem
119
Effective Bounds for Systems
126
Analytic Elements
128
Some Transfer Theorems
133
Logarithms
138
The Binomial Series
140
The Hypergeometric Function of Euler and Gauss
150
Effective Bounds Singular Disks 1 The DworkFrobenius Theorem
155

Totally Ramified Extensions
33
Zeta Functions 1 Logarithms
38
Newton Polygons for Power Series
41
Newton Polygons for Laurent Series
46
The Binomial and Exponential Series
49
Diendonnes Theorem
53
S Analytic Representation of Additive Characters
56
Meromorphy of the Zeta Function of a Variety
61
Condition for Rationality
71
Rationality of the Zeta Function
74
Appendix to Chapter II
76
Differential Equations 1 Differential Equations in Charateristic p
77
Nilpotent Differential Operators KatzHonda Theorem
81
Differential Systems
86
The Theorem of the Cyclic Vector
89
The Generic Disk Radius of Convergence
92
Global Nilpotence Katzs Theorem
98
Regular Singularities Fuchs Theorem
100
Formal Fuchsian Theory
102
Effective Bounds Ordinary Disks 1 padic Analytic Functions
114
the Case of Nilpotent Monodromy The Christol Dwork Theorem Outline of the Proof
159
Proof of Step V
168
Proof of Step IV The Shearing Transformation
169
Proof of Step III Removing Apparent Singularities
170
The Operators 4 and tl
173
Proof of Step I Construction of Frobenius
176
Proof of Step II Effective Form of the Cyclic Vector
180
Effective Bounds The Case of Unipotent Monodromy
189
Transfer Theorems into Disks with One Singularity 1 The Type of a Number
199
a First Estimate
203
The Theorem of Transfer of Radii of Convergence
212
Differential Equations of Arithmetic Type 1 The Height
222
The Theorem of BombieriAndre
226
Transfer Theorems for Differential Equations of Arithmetic Type
234
GSeries The Theorem of Chudnovsky
263
Convergence Polygon for Differential
301
Archimedean Estimates
307
bibliography
317
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