Homomorphisms and Duality

Abstract

This chapter describes the elementary theory of homomorphisms and endomorphisms of an abelian manifold. First we relate the rational and complex representations to a purely algebraic representation on the points of finite order. Then we prove the complete reducibility theorem of Poincaré, showing that an abelian manifold admits a product decomposition into simple ones, up to isogeny. Finally, we deal with the duality which arises from the nondegenerate hermitian form, and show how the dual manifold corresponds to divisor classes of divisors algebraically equivalent to 0. The duality includes an essentially algebraic pairing between points and such divisors, and a formula in the last section relates this algebraic pairing with the analytic data and the Riemann form.