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Number theory and geometry. An introduction to arithmetic geometry. (English) Zbl 1432.11002

Pure and Applied Undergraduate Texts 35. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-5016-8/hbk; 978-1-4704-5190-5/ebook). xv, 488 p. (2019).
The book main emphasis is on describing the solutions of simple Diophantine equations as “geometric objects ” over different “arithmetic” subsets of the real numbers as a manner to introduce the basics of elementary number theory. Detailed examples appear throughout, as well as nice black and white pictures mainly of famous number theorists. There are many exercises at the end of each chapter, but no solutions (or hints for the more difficult) for some of them, like, e.g., odd numbered ones as in classical Kenneth Rosen’s book. The treatment of quadratic Diophantine equations is done in more detail than in Loo-Keng Hua’s book (that is is missing in the bibliography). The most important contribution of the author is a detailed introduction for beginners of elliptic curves that is simpler than in Anthony Knapp’s book and more detailed than in classic Tate-Silverman’s book. The book is an introduction to a more advanced book of the author on the same subject.
A flavor of the contents is: (a) Chapter 1 Introduction roots of polynomials, lines, quadratic equations and conic sections, cubic equations and elliptic curves, curves of higher degree, Diophantine equations, Hilbert’s tenth problem, exercises. Part 1. Integers, polynomials, lines and congruences. Part 2. Quadratic congruences and quadratic equations. Part 3. Cubic equations and elliptic curves.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11D09 Quadratic and bilinear Diophantine equations
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
97F60 Number theory (educational aspects)
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