Complex Multiplication

  • Serge Lang

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 255)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Serge Lang
    Pages 1-34
  3. Serge Lang
    Pages 53-83
  4. Serge Lang
    Pages 84-121
  5. Serge Lang
    Pages 148-162
  6. Serge Lang
    Pages 163-178
  7. Back Matter
    Pages 179-184

About this book


The small book by Shimura-Taniyama on the subject of complex multi­ is a classic. It gives the results obtained by them (and some by Weil) plication in the higher dimensional case, generalizing in a non-trivial way the method of Deuring for elliptic curves, by reduction mod p. Partly through the work of Shimura himself (cf. [Sh 1] [Sh 2], and [Sh 5]), and some others (Serre, Tate, Kubota, Ribet, Deligne etc.) it is possible today to make a more snappy and extensive presentation of the fundamental results than was possible in 1961. Several persons have found my lecture notes on this subject useful to them, and so I have decided to publish this short book to make them more widely available. Readers acquainted with the standard theory of abelian varieties, and who wish to get rapidly an idea of the fundamental facts of complex multi­ plication, are advised to look first at the two main theorems, Chapter 3, §6 and Chapter 4, §1, as well as the rest of Chapter 4. The applications of Chapter 6 could also be profitably read early. I am much indebted to N. Schappacher for a careful reading of the manu­ script resulting in a number of useful suggestions. S. LANG Contents CHAPTER 1 Analytic Complex Multiplication 4 I. Positive Definite Involutions . . . 6 2. CM Types and Subfields. . . . . 8 3. Application to Abelian Manifolds. 4. Construction of Abelian Manifolds with CM 14 21 5. Reflex of a CM Type . . . . .


Abelian varieties Abelian variety Finite Komplexe Multiplikation algebra congruence construction elliptic curve function functions manifold presentation theorem transformation zeta function

Authors and affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York 1983
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-5487-4
  • Online ISBN 978-1-4612-5485-0
  • Series Print ISSN 0072-7830
  • About this book