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Number theory. Volume II: Analytic and modern tools. (English) Zbl 1119.11002
Graduate Texts in Mathematics 240. New York, NY: Springer (ISBN 978-0-387-49893-5/hbk). xiii, 596 p. EUR 46.95/net; SFR 77.00; \$ 59.95; £ 36.00 (2007).

The second volume of Cohen’s book (for the first volume, see (2007; Zbl 1119.11001)) also contains two parts; the first deals with arithmetic aspects of Dirichlet series and $L$-series: it starts with a rather long preliminary Chapter 9 on Bernoulli numbers and polynomials as well as the Gamma function, then discusses $L$-series for number fields in Chapter 10 (keywords include special values, Epstein’s zeta function, Kronecker’s limit formula, and the prime number theorem), and finally treats the $p$-adic theory (including the Gross-Koblitz formula for Gauss sums) in Chapter 11.

The fourth and last part is called “Modern Tools”; its chapters were written by leading specialists in the field, and contain only a few exercises. Chapter 12 discusses applications of linear forms in logarithms and was written by Y. Bugeaud, G. Hanrot and M. Mignotte, who recently combined this technique with that of modular forms to solve the problem of finding all perfect powers in the Fibonacci sequence; here they also explain how to attack simultaneous Pell equations, Thue equations, and Catalan’s equation. Chapter 13 on rational points on curves of higher genus is written by S. Duquesne; it briefly outlines the basic background and explains methods of Chabauty type. Chapter 14 on the super-Fermat equation was written by the author and explains the recent results on the Diophantine equation ${x}^{p}+{y}^{q}+{z}^{r}=0$ for integers $p,q,r\ge 2$. The next chapter on the modular approach to Diophantine equations (Ribet’s level-lowering theorem and Wiles’ mdularity theorem) was provided by S. Siksek. These three chapters are mainly expositional and do contain only few proofs. The last chapter 16, on the other hand, contains the full proof of Catalan’s conjecture due to Mihailescu.

The general remarks made in the review of volume I also apply here. Both volumes contain a wealth of information on recent work and give a lucid description of the methods currently used in Diophantine analysis.

##### MSC:
 11-01 Textbooks (number theory) 11-02 Research monographs (number theory) 11Mxx Analytic theory of zeta and $L$-functions 11Sxx Algebraic number theory: local and $p$-adic fields 11Dxx Diophantine equations 11Gxx Arithmetic algebraic geometry (Diophantine geometry) 11B68 Bernoulli and Euler numbers and polynomials 11J86 Linear forms in logarithms; Baker’s method