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Number theory. Volume I: Tools and Diophantine equations. (English) Zbl 1119.11001
Graduate Texts in Mathematics 239. New York, NY: Springer (ISBN 978-0-387-49922-2/hbk). xxiii, 650 p. (2007).
Although the interest in Diophantine problems has fueled a lot of research in algebraic number theory and arithmetic geometry, textbooks dedicated to general Diophantine analysis (as opposed to books on problems of a special type, such as elliptic curves or Fermat’s Last Theorem) are rare. The few books that come to mind are R. D. Carmichael’s, Diophantine Analysis [New York: John Wiley and Sons. VI (1915; JFM 45.0283.11)], L. J. Mordell’s famous “Diophantine Equations” [Pure and Applied Mathematics, 30. London-New York: Academic Press (1969; Zbl 0188.34503)] and S. Lang’s Diophantine Geometry [Interscience Tracs in Pure and Applied Mathematics. 11. New York and London: Interscience Publishers, a division of John Wiley and Sons. (1962; Zbl 0115.38701)].
The book under review deals with Diophantine analysis from a number-theoretic point of view. Its author shares Mordell’s taste for concrete Diophantine problems, but wisely avoids to follow the latter’s concept: Mordell’s “classification” of Diophantine problems was already outdated when his book appeared. In fact, the appeal of a given problem is usually measured by the techniques created for solving it, and it is therefore only consequent to give an exposition oriented towards the tools rather than the problems.
The first volume starts with a brief (historical) introduction to Diophantine equations, and then presents the basic tools of the trade, mostly with proofs. Chapter 2 introduces residue classes, quadratic reciprocity, lattices and LLL-reduction, finite fields, Gauss and Jacobi sums, and the Weil bounds. In Chapter 3, Cohen reviews algebraic number theory, with an emphasis on cyclotomic fields and Stickelberger’s theorem. Chapters 4 and 5 give the basic theory of \(p\)-adic number fields and their extensions, and the theory of quadratic forms from the viewpoint of Local-Global Principles.
The second part of volume I deals with Diophantine equations: in Chapter 6, problems of degree \(\leq 4\) as well as Fermat’s Last Theorem are discussed. Chapter 7 provides the relevant results from the theory of elliptic curves, and Chapter 8 discusses Diophantine aspects of elliptic curves, namely descent, \(L\)-series, Heegner points, and integral points via elliptic logarithms.
It should be clear from this brief description of the content that the author’s aim is not primarily the algorithmic aspect of the solution of Diophantine equations (this is discussed in detail in N. P. Smart’s book [The algorithmic resolution of Diophantine equations. London Mathematical Society Student Texts. 41. Cambridge: Cambridge University Press (1998; Zbl 0907.11001)]) but rather the mathematics that lies behind some of the most spectacular results of the last few years, in particular Fermat’s Last Theorem and Catalan’s equation. Each chapter ends with exercises, ranging from simple to quite challenging problems. The clarity of the exposition is the one we expect from the author of two highly successful books on computational number theory [Zbl 0786.11071; Zbl 0977.11056], and makes this volume a must-read for researchers in Diophantine analysis.

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11Rxx Algebraic number theory: global fields
11Sxx Algebraic number theory: local fields
11Dxx Diophantine equations
11G05 Elliptic curves over global fields
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