Analytic Complex Multiplication

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 formE 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.