This book is an outgrowth of a course, taught by the author at MIT during fall 2007, on p-adic ordinary differential equations. The target audience was graduate students with some prior background in algebraic number theory, including exposure to p-adic numbers, but not necessarily with any background in p-adic analytic geometry (of either the Tate or Berkovich flavors).

Custom would dictate that ordinarily this preface would continue with an explanation of what p-adic differential equations are, and why they matter. Since we have included a whole chapter on this topic (Chapter 0), we will devote this preface instead to a discussion of the origin of the book, its general structure, and what makes it different from previous books on the subject.

The subject of p-adic differential equations has been treated in several previous books. Two that we used in preparing the MIT course, and to which we make frequent reference in the text, are those of Dwork, Gerotto, and Sullivan [80] and of Christol [42]. Another existing book is that of Dwork [78], but it is not a general treatise; rather, it focuses in detail on hypergeometric functions.

However, this book develops the theory of p-adic differential equations in a manner that differs significantly from most prior literature. Key differences include the following.

• We limit our use of cyclic vectors. This requires an initial investment in the study of matrix inequalities (Chapter 4) and lattice approximation arguments (especially Lemma 8.6.1), but it pays off in significantly stronger results.

• We introduce the notion of a Frobenius descendant (Chapter 10). This complements the older construction of Frobenius antecedents, particularly in dealing with certain boundary cases where the antecedent method does not apply.