Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties.

摘要

The author's aim in these notes is to give an introduction to the theory of Abelian varieties. At many places, they will not give detailed proofs, but will try to discuss the 'concrete' origins of the theory. Therefore they will spend most of these lectures explaining basic results about Riemann surfaces and algebraic curves and about the Abelian varieties which are associated to them, the Jacobian varieties. The content of these notes is quite classical: most of it is known since the beginning of the century. They follow a rather geometrical and analytical approach to the subject, in order to stay 'as concrete as possible'; the arithmetical aspects will only be mentioned at the very end of the lectures. They have tried to use a language familiar to theoretical physicists: the only prerequisites are some familiarity with the basic facts concerning holomorphic functions, differential forms and manifolds. These notes originated in lectures given at the meeting 'Number Theory and Physics', held at the 'Centre de Physique', Les Houches, France, March 7-16, 1989. The present notes are an extended version of part of the material included in the lectures.