##
**\(p\)-adic differential equations.**
*(English)*
Zbl 1213.12009

Cambridge Studies in Advanced Mathematics 125. Cambridge: Cambridge University Press (ISBN 978-0-521-76879-5/hbk). xvii, 380 p. (2010).

The theory of \(p\)-adic differential equations is a relatively recent branch of mathematics. The first paper (containing a Cauchy type local existence theorem) was published in 1937 [E. Lutz, J. Reine Angew. Math. 177, 238–247 (1937; Zbl 0017.05307)], while systematic studies were initiated by Dwork in the 1960s. From the very beginning, these studies were motivated by applications to number theory and algebraic geometry. Two previous books on this subject mention such applications already in their titles: B. Dwork, G. Gerotto and F. J. Sullivan, An introduction to \(G\)-functions. Annals of Mathematics Studies. 133. Princeton, NJ: Princeton University Press (1994; Zbl 0830.12004); P. Robba and G. Christol, Équations différentielles \(p\)-adiques. Applications aux sommes exponentielles. Paris: Hermann (1994; Zbl 0868.12006).

The techniques used in this theory are different from those employed by the classical theory of differential equations; that is dictated by the subject itself, primarily by a much more complicated algebraic structure of the field \(\mathbb Q_p\) possessing finite algebraic extensions of all degrees. The results are often very different too – the solutions of simplest equations without singularities (like the one for the exponential function) exist only on certain finite disks. The most interesting results are connected with overconvergence phenomena where solutions of some equations behave in this sense better than expected. Usually such equations “come from geometry”, such as the Picard-Fuchs equations, which arise from integrals of algebraic functions. Among related research fields, one can mention the theory of \(p\)-adic \(q\)-difference equations and the theory of (Carlitz-type) differential equations over fields of positive characteristic [see L. di Vizio, “Introduction to \(p\)-adic \(q\)-difference equations (weak Frobenius structure and transfer theorems)”, in: Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter, 615–675 (2004; Zbl 1070.12004); A. N. Kochubei, Analysis in positive characteristic. Cambridge Tracts in Mathematics 178. Cambridge: Cambridge University Press (2009; Zbl 1171.12005)].

The book by Kedlaya is devoted completely to \(p\)-adic differential equations as an independent subject. It consists essentially of six parts. Their description in the preface to the book is as follows.

“Part I is preliminary, collecting some basic tools of \(p\)-adic analysis. However, it also includes some facts of matrix analysis (the study of the variation of numerical invariants attached to matrices as a function of the matrix entries) which may not be familiar to the typical reader.

Part II introduces some formalism of differential algebra, such as differential rings and modules, twisted polynomials, and cyclic vectors, and applies these to fields equipped with a nonarchimedean norm.

Part III begins with the study of \(p\)-adic differential equations in earnest, developing some basic theory for differential modules on rings and annuli, including the Christol-Dwork theory of variation of the generic radius of convergence and the Christol-Mebkhout decomposition theory. We also include a treatment of \(p\)-adic exponents, culminating in the Christol-Mebkhout structure theorem for \(p\)-adic differential modules on an annulus satisfying the Robba condition (i.e. having intrinsic generic radius of convergence everywhere equal to 1).

Part IV introduces some formalism of differential algebra and presents (without full proofs) the theory of slope filtrations for Frobenius modules over the Robba ring.

Part V introduces the concept of a Frobenius structure on a \(p\)-adic differential module, to the point of stating the \(p\)-adic local monodromy theorem and sketching briefly the proof techniques using either \(p\)-adic exponents or Frobenius slope filtrations. We also discuss effective convergence bounds for solutions of \(p\)-adic differential equations.

Part VI consists of a series of brief discussions of several areas of applications of the theory of \(p\)-adic differential equations. These are somewhat more didactic, and much less formal, than in the other parts; they are meant primarily as suggestions for further reading.”

As a whole, the book is a valuable source of information both for specialists in the field and those who approach the subject from various directions, such as number theory, algebraic geometry, or classical differential equations.

The techniques used in this theory are different from those employed by the classical theory of differential equations; that is dictated by the subject itself, primarily by a much more complicated algebraic structure of the field \(\mathbb Q_p\) possessing finite algebraic extensions of all degrees. The results are often very different too – the solutions of simplest equations without singularities (like the one for the exponential function) exist only on certain finite disks. The most interesting results are connected with overconvergence phenomena where solutions of some equations behave in this sense better than expected. Usually such equations “come from geometry”, such as the Picard-Fuchs equations, which arise from integrals of algebraic functions. Among related research fields, one can mention the theory of \(p\)-adic \(q\)-difference equations and the theory of (Carlitz-type) differential equations over fields of positive characteristic [see L. di Vizio, “Introduction to \(p\)-adic \(q\)-difference equations (weak Frobenius structure and transfer theorems)”, in: Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter, 615–675 (2004; Zbl 1070.12004); A. N. Kochubei, Analysis in positive characteristic. Cambridge Tracts in Mathematics 178. Cambridge: Cambridge University Press (2009; Zbl 1171.12005)].

The book by Kedlaya is devoted completely to \(p\)-adic differential equations as an independent subject. It consists essentially of six parts. Their description in the preface to the book is as follows.

“Part I is preliminary, collecting some basic tools of \(p\)-adic analysis. However, it also includes some facts of matrix analysis (the study of the variation of numerical invariants attached to matrices as a function of the matrix entries) which may not be familiar to the typical reader.

Part II introduces some formalism of differential algebra, such as differential rings and modules, twisted polynomials, and cyclic vectors, and applies these to fields equipped with a nonarchimedean norm.

Part III begins with the study of \(p\)-adic differential equations in earnest, developing some basic theory for differential modules on rings and annuli, including the Christol-Dwork theory of variation of the generic radius of convergence and the Christol-Mebkhout decomposition theory. We also include a treatment of \(p\)-adic exponents, culminating in the Christol-Mebkhout structure theorem for \(p\)-adic differential modules on an annulus satisfying the Robba condition (i.e. having intrinsic generic radius of convergence everywhere equal to 1).

Part IV introduces some formalism of differential algebra and presents (without full proofs) the theory of slope filtrations for Frobenius modules over the Robba ring.

Part V introduces the concept of a Frobenius structure on a \(p\)-adic differential module, to the point of stating the \(p\)-adic local monodromy theorem and sketching briefly the proof techniques using either \(p\)-adic exponents or Frobenius slope filtrations. We also discuss effective convergence bounds for solutions of \(p\)-adic differential equations.

Part VI consists of a series of brief discussions of several areas of applications of the theory of \(p\)-adic differential equations. These are somewhat more didactic, and much less formal, than in the other parts; they are meant primarily as suggestions for further reading.”

As a whole, the book is a valuable source of information both for specialists in the field and those who approach the subject from various directions, such as number theory, algebraic geometry, or classical differential equations.

Reviewer: Anatoly N. Kochubei (Kyïv)

### MSC:

12H25 | \(p\)-adic differential equations |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

12H05 | Differential algebra |

14G20 | Local ground fields in algebraic geometry |

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |