I. Complex Manifolds and Vector Bundles.- § 1. Complex Manifolds and the Example of the Complex Tori.- § 2. Complex Analytic Vector Bundles and the Example of Line Bundles of Divisors.- § 3. Factors of Automorphy and Complex Analytic Vector Bundles.- II. Riemann Surfaces.- § 4. Markings of Riemann Surfaces and Characteristic Classes of Factors of Automorphy.- § 5. Abelian Differentials and the Jacobi Variety of a Riemann Surface.- § 6. Meromorphic Abelian Differentials and the Prime Function of a Riemann Surface.- III. Generalized Theta Functions.- § 7. Theta Factors of Automorphy and Generalized Theta Functions.- § 8. Generalized Theta Functions and Canonical Subvarieties of the Jacobi Variety.- § 9. Relations Between Theta Factors of Automorphy.- § 10. Dimensions of Spaces of Generalized Theta Functions.- § 11. Induced Theta Factors and Theta Functions on Riemann Surfaces.- IV. Prym Differentials.- § 12. Prym Differentials and Generalized Theta Functions.- § 13. Periods and the Period Matrix for Prym Differentials.- § 14. The Riemann Equality for Prym Periods.- § 15. Regular Prym Differentials.- Appendix. Some Topics in the Classical Theory of Theta Functions.- § 16. Classification of Scalar Factors of Automorphy for Complex Tori.- § 17. Relatively Automorphic Functions: the Theta Series.- § 18. Jacobi Varieties: Abelian and Riemannian Theta Functions.- § 19. Some Analytic Cohomology Groups for Complex Tori.- References.- Index of Theorems.- Index of Notation.

The investigation of the relationships between compact Riemann surfaces (al­ gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper­ 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M.