zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 {\it Y. Bugeaud, G. Hanrot} and {\it 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 {\it 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 {\it 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.

11-01Textbooks (number theory)
11-02Research monographs (number theory)
11MxxAnalytic theory of zeta and $L$-functions
11SxxAlgebraic number theory: local and $p$-adic fields
11DxxDiophantine equations
11GxxArithmetic algebraic geometry (Diophantine geometry)
11B68Bernoulli and Euler numbers and polynomials
11J86Linear forms in logarithms; Baker’s method
Full Text: DOI