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)
A classical introduction to modern number theory. 2nd ed. (English) Zbl 0712.11001
Graduate Texts in Mathematics, 84. New York etc.: Springer-Verlag. xiv, 389 p. DM 98.00 (1990).
This is a somewhat expanded version of the first edition that appeared in 1982 [Springer Graduate texts in mathematics 84, (1982; Zbl 0482.10001). The first edition of 1982 consisted of 18 chapters which have been included in the second edition without alterations. The second edition contains two new chapters, one on the Mordell-Weil theorem for elliptic curves and one on recent developments in arithmetic geometry. The authors wrote the first edition with the purpose to give insight into modern developments in number theory by showing their close relationship with classical, 19th century number theory. The authors wrote the new chapters 19 and 20 in the same spirit. In chapter 19, they give a proof of the Mordell-Weil theorem for elliptic curves over without using Kummer theory or algebraic geometry: they first give Cassels’ proof of the weak Mordell-Weil theorem which uses only a weaker version of Dirichlet’s unit theorem for number fields; and then they derive the Mordell-Weil theorem using the standard descent argument which is worked out by elementary arithmetic. Chapter 19 is meant as a preparation for chapter 20, in which the authors give a very interesting overview of the important developments in arithmetic geometry after the appearance of the first edition of their book. Among other things, they discuss the Mordell conjecture proved by Faltings, the Taniyama-Weil conjecture and the result of Frey, Serre and Ribet that this implies Fermat’s last theorem, recent progress on the Birch-Swinnerton-Dyer conjecture by Coates-Wiles, Gross-Zagier, Rubin, and Kolyvagin, and the derivation of Gauss’ class number conjecture from the results of Gross-Zagier. In chapter 20, the authors do not give proofs but they give sufficient background to understand and appreciate the results. Chapter 20 is an excellent introduction for those who want to study the subject more thoroughly.
Reviewer: J.-H.Evertse

11-01Textbooks (number theory)
11AxxElementary number theory
11GxxArithmetic algebraic geometry (Diophantine geometry)
11NxxMultiplicative number theory
11RxxAlgebraic number theory: global fields
11TxxFinite fields and finite commutative rings (number-theoretic)
11DxxDiophantine equations
11G05Elliptic curves over global fields
11G40L-functions of varieties over global fields
11D41Higher degree diophantine equations
14H52Elliptic curves