The study of arithmetic differential operators is a novel and promising area of mathematics. This complete introduction to the subject starts with the basics: a discussion of p-adic numbers and some of the classical differential analysis on the field of p-adic numbers leading to the definition of arithmetic differential operators on this field. Buium's theory of arithmetic jet spaces is then developed succinctly in order to define arithmetic operators in general. Features of the book include a comparison of the behaviour of these operators over the p-adic integers and their behaviour over the unramified completion, and a discussion of the relationship between characteristic functions of p-adic discs and arithmetic differential operators that disappears as soon as a single root of unity is adjoined to the p-adic integers. This book is essential reading for researchers and graduate students who want a first introduction to arithmetic differential operators over the p-adic integers.
• The first introductory book on the subject • Accessible to those with a grasp of algebraic number theory • Allows those without a strong training in commutative algebra and algebraic geometry to achieve a deep understanding of the subject
Contents
1. Introduction; 2. The p-adic numbers Q_p; 3. Some classical analysis on Q_p; 4. Analytic functions on Z_p; 5. Arithmetic differential operators on Z_p; 6. A general view of arithmetic differential operators; 7. Analyticity of arithmetic differential operators; 8. Characteristic functions: standard p-adic coordinates; 9. Characteristic functions: harmonic arithmetic coordinates; 10. Differences between arithmetic differential operators over Z_p and Z_p^{unr}; References.
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