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Analytic Complex Multiplication

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

Abstract

This chapter is essentially elementary, and lays the foundations for the study of the endomorphisms of complex toruses known as complex multiplications. Let V be a vector space of dimension n over the complex numbers. Let Λ be a lattice in V. The quotient complex analytic group V/Λ is called a complex torus. We assume known the basic facts concerning Riemann forms and the projective embedding of such toruses. A (non-degenerate) Riemann form E on V/Λ is an alternating non-degenerate form on V such that E(x, y) ∊ Z for x, y ∊ Λ, and such that the form E(ix, y) is symmetric positive definite. Equivalently, one may say that E is the imaginary part of a positive definite hermitian form on V, and takes integral values on Λ. The torus admits a projective embedding if and only if it admits a Riemann form, and such a projective embedding is obtained by projective coordinates given by theta functions. We shall not need to know anything about such theta functions aside from their existence. An abelian manifold is a complex torus which admits a Riemann form.

Keywords

Abelian Variety Division Algebra Rational Representation Galois Extension Admissible Pair 
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Copyright information

© Springer-Verlag New York Inc. 1983

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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