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**A course in \(p\)-adic analysis.**
*(English)*
Zbl 0947.11035

Graduate Texts in Mathematics. 198. New York, NY: Springer. xv, 437 p. (2000).

The first five chapters of this book form an excellent first excursion to the ‘\(p\)-adic realm’. Only standard knowledge of calculus, algebra and point set topology is required. The basic theory of \(p\)-adic fields, continuity and differentiability is developed (e.g. \(\mathbb{Z}_p\), \(\mathbb{Q}_p\), Hensel’s lemma and Newton approximation, roots of unity, each automorphism of \(\mathbb{Q}_p\) is the identity, ultrametric spaces, valuations, equivalence, all valuations on \(\mathbb{Q}\), finite-dimensional normed spaces, extensions of valuations, ramification, Eisenstein polynomials, classification of locally compact fields, algebraic closure of \(\mathbb{Q}_p\), the field \(\mathbb{C}_p\) of the \(p\)-adic complex numbers, continuous functions on \(\mathbb{Z}_p\), convolution, Mahler series and its generalizations by van Hamme, locally constant functions, van der Put expansion, generating functions, Bell polynomials, (strict) differentiability of orders 1 and 2, differentiation of Mahler series, restricted formal power series, Gauss norms, exp and log, Iwasawa logarithm, Volkenborn integral, Bernoulli numbers, Clausen-von Staudt theorem). Noteworthy is the attention to linear and Euclidean models yielding nice pictures of fractals enabling the reader to ‘visualize’ \(\mathbb{Z}_p\), and the study of the \(p\)-adic solenoid.

Particularly elegant is the proof of the existence of an extension of a valuation by using the concept of a generalized absolute value \((|x+y|\leq C\max (|x|,|y|)\) where \(C\) may be \(>1\): it runs so smoothly!

Other new features are the mean value and Rolle’s theorem, formulated for restricted power series, and Diarra’s construction of a spherically complete extension of \(\mathbb{C}_p\) by means of ultrafilters. Some new – to the point – terminology and catch phrases are proposed (such as ‘dressed’ and ‘stripped’ balls for ‘closed’ and ‘open’ balls in an ultrametric space and ‘the strongest wins’ for ‘\(|x|>|y|\Rightarrow |x+y|= |x|\)’).

Chapter 6 treats the basic theory of analytic functions and elements. This part will particularly be welcomed by many readers as so far no such introduction on an elementary level exists. The topics are power series (a careful distinction between a formal power series and its corresponding function on some disk is always maintained), Newton polygons, Laurent series, zeros of power series, entire functions, maximum principle, Hadamard’s three-circle theorem, Weierstrass products, rational functions, Mittag-Leffler decompositions, Motzkin factorizations, infraconnected sets, analytic elements, Amice-Fresnel theorem, Christol-Robba theorem.

In the final Chapter 7 the theory is applied to define special functions (e.g. \(p\)-adic Gamma function, Artin-Hasse exponential, Dwork exponential) and to derive interesting congruences from it. Also Gauss sums and the Gross-Koblitz formula are treated.

This well-written book, complete with all proofs and a wealth of exercises, is perfectly suited as a text book for introductory courses.

Particularly elegant is the proof of the existence of an extension of a valuation by using the concept of a generalized absolute value \((|x+y|\leq C\max (|x|,|y|)\) where \(C\) may be \(>1\): it runs so smoothly!

Other new features are the mean value and Rolle’s theorem, formulated for restricted power series, and Diarra’s construction of a spherically complete extension of \(\mathbb{C}_p\) by means of ultrafilters. Some new – to the point – terminology and catch phrases are proposed (such as ‘dressed’ and ‘stripped’ balls for ‘closed’ and ‘open’ balls in an ultrametric space and ‘the strongest wins’ for ‘\(|x|>|y|\Rightarrow |x+y|= |x|\)’).

Chapter 6 treats the basic theory of analytic functions and elements. This part will particularly be welcomed by many readers as so far no such introduction on an elementary level exists. The topics are power series (a careful distinction between a formal power series and its corresponding function on some disk is always maintained), Newton polygons, Laurent series, zeros of power series, entire functions, maximum principle, Hadamard’s three-circle theorem, Weierstrass products, rational functions, Mittag-Leffler decompositions, Motzkin factorizations, infraconnected sets, analytic elements, Amice-Fresnel theorem, Christol-Robba theorem.

In the final Chapter 7 the theory is applied to define special functions (e.g. \(p\)-adic Gamma function, Artin-Hasse exponential, Dwork exponential) and to derive interesting congruences from it. Also Gauss sums and the Gross-Koblitz formula are treated.

This well-written book, complete with all proofs and a wealth of exercises, is perfectly suited as a text book for introductory courses.

Reviewer: Wim Schikhof (Nijmegen)

### MSC:

11Sxx | Algebraic number theory: local fields |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

30G06 | Non-Archimedean function theory |

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |