zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
The arithmetic of elliptic curves. (English) Zbl 0585.14026
Graduate Texts in Mathematics, 106. New York etc.: Springer-Verlag. XII, 400 p. DM 148.00 (1986).

This book provide a nice introduction to a presently very active part of mathematics.

Its first part gives the fundamental notions, beginning by short chapters reviewing the main definitions and results for algebraic varieties, algebraic curves (including Riemann-Roch theorem); it goes on with the basic definitions for elliptic curves (Weierstraß models, the group law, isogenies, Tate module...) and their formal group.

The chapter V studies elliptic curves on finite fields; after proving the Weil conjectures for them, it discusses the Hasse invariant and supersingular curves. Chapter VI studies elliptic curves over using the Weierstraß -function; the structure of the group of torsion is given. The chapter ends by a discussion of the Lefschetz principle allowing to extend results to other fields of characteristics 0. The chapter VII studies elliptic curves over a local field: it discusses the minimal model, and gives the Néron-Ogg-Shafarevich criterion for good reduction. With the chapter VIII begins the theory of elliptic curves over a global field. Its purpose is to prove the Mordell- Weil theorem (The group of rational points of E over K is of finite type) using descent theory and heights. The case where K= is settled in first by down to earth considerations. Mestre’s example of an elliptic curve of rank 12 is given. Chapter IX studies the integral points using Siegel’s theorem. It also discusses effective problems with the help of Baker’s theory. It proves the Shafarevich theorem about the finiteness of the number of elliptic curves with good reduction outside a given set of primes. Chapter X (the summit of the book) discusses descent theory to compute the Mordell-Weil group. It defines the Selmer group and the Shafarevich groups. It ends with a discussion about the famous family y 2 =x 3 -4dx·

The book ends with 3 appendices; the first is about characteristics 2 and 3, the second is about Galois theory, the third is a brief summary of other important questions (the contents of a second volume ?). Note that this book has a good index and gives an important bibliography.

I think that this book will be very useful for people beginning in the field of research about elliptic curves.

Reviewer: R. Gillard

MSC:
11G05Elliptic curves over global fields
11-01Textbooks (number theory)
11G07Elliptic curves over local fields
11G20Curves over finite and local fields
11G40L-functions of varieties over global fields
14H45Special curves and curves of low genus
14G05Rational points
14H25Arithmetic ground fields (curves)
14-01Textbooks (algebraic geometry)
14H52Elliptic curves
11D25Cubic and quartic diophantine equations