## An Introduction to G-functionsWritten for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of After presenting a review of valuation theory and elementary |

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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 |