## p-adic Differential EquationsOver the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study. |

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### Contents

1 | |

Part I Tools of padic Analysis | 11 |

Part II Differential Algebra | 75 |

Part III padic Differential Equations on Discs and Annuli | 133 |

Part IV Difference Algebra and Frobenius Modules | 241 |

Part V Frobenius Structures | 289 |

Part VI Areas of Application | 345 |

### Common terms and phrases

algebra analogue annulus archimedean basis bounded Chapter choose Christol coefﬁcients cohomology compute Corollary cyclic vector cyclic vector theorem decomposition theorem deduce deﬁned Deﬁnition discretely valued dualizable Dwork eigenvalues element equipped equivalent étale étale cohomology Example exercise exists exponents ﬁeld ﬁnite differential module ﬁnite extension ﬁnite free ﬁrst free module Frobenius lift Frobenius structure function Galois Galois representations Hodge horizontal sections hypothesis implies IR(V isomorphism Kedlaya Let F Math matrix of action module of rank module over F monodromy theorem morphism Newton polygon Newton slopes nilpotent nonarchimedean differential nonzero Note p-adic differential equations p-adic Hodge theory positive integer proof of Theorem Proposition Prove pure of norm quotient radius of convergence ramiﬁed reﬁned Remark residue ﬁeld result satisﬁes seminorm sequence singular values slope ﬁltration spectral radius spherically complete submodule subsidiary radii sufﬁces Suppose supremum norm theory trivial unipotent unique unramiﬁed