Complex Multiplication pp 1-34 | Cite as

# Analytic Complex Multiplication

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## 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## Preview

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