Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0911.14015
Silverman, Joseph H.
Advanced topics in the arithmetic of elliptic curves. (English)
[B] Graduate Texts in Mathematics. 151. New York, NY: Springer-Verlag. xiii, 525 p. DM 98.00; öS 764.40; sFr. 98.00 (1994). ISBN 0-387-94325-0/hbk

The monograph is a continuation of the same author's book: {\it J. Silverman}, ``The arithmetic of elliptic curves'' (1986; Zbl 0585.14026). However it can be used independently from this book, on the basis of some familiarity with the basic theory of elliptic curves.\par The first chapter treats elliptic and modular functions on the basis of classical complex analysis. Jacobi's product formula for $\Delta (\tau)$, the action of Hecke-operators on modular forms and the $L$-series attached to modular forms are included. The second chapter is ``Complex multiplication'', including a brief review of class field theory and ends with a detailed study of the $L$-series attached to a $C\text {M}$ elliptic curve. The third chapter is about elliptic surfaces. It includes a brief introduction to the geometry of algebraic surfaces (intersection theory and minimal models) (not with complete proofs). It culminates in the proof of the Mordell-Weil theorem for function fields, construction of the canonical height, and specialization theorems for the canonical height. The fourth chapter is a treatment of the Néron-model. In a sense this is a continuation of the previous chapter, based on the language of schemes, since it treats the case of arithmetic surfaces. It gives a definition of the Néron-model, and studies the fibres (Kodaira-Néron classification). It also includes an introduction to intersection theory on arithmetic surfaces. Also, Ogg's formula on the relation between the conductor, the minimal discriminant, and the special fibre of the minimal model (for a local field) is verified. The last two chapters treat topics for curves over local fields, namely the $p$-adic uniformization, the Tate-curve and Serre's proof for the integrality of the $j$-invariant of a CM-curve (chapter V), and local height functions (chapter VI).\par The book is well written. It covers a large part of a fascinating and advanced development in mathematics, combining (complex) analysis, geometry and number theory. It keeps the needed advanced technology on a level absolutely necessary for the treatment of the subject but well enough explained for the actual purposes. Thus the book could be recommended for a wide audience to become acquainted with this fascinating topic.
[H.Kurke (Berlin)]

MSC 2000:: *14H52 Elliptic curves
14G40 Arithmetic varieties and schemes
11-02 Research monographs (number theory)
11Gxx Arithmetic algebraic geometry (Diophantine geometry)
14-02 Research monographs (algebraic geometry)
11G07 Elliptic curves over local fields
11G05 Elliptic curves over global fields
14G10 Zeta-functions and related questions

Keywords: elliptic functions; $L$-series; complex multiplication; $p$-adic uniformization; modular functions; Mordell-Weil theorem for function fields; canonical height; Néron-model; minimal model

Citations: Zbl 0585.14026

Cited in: Zbl 1105.11016 Zbl 1046.11036 Zbl 0973.11062

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]