there are a couple of relevant paragraphs in Basic algebraic geometry by shafarevich which i recommend, as well as a historical appendix. e.g. he gives an example to show that unlike the one dimensional case, there exist lattices in C^2 for which not even any meromorphic functions can be periodic (chapter VIII.1). in addition to siegel and swinnerton dyer, mumford's other theta functions book is one i would recommend, volume 1 of the three volume series, something like "tata lectures on theta I". Also in lecture IV of his Curves and their Jacobians, now included in his "red book", at least the first 8 pages give an invaluable and succinct introduction to principally polarized abelian varieties and their "moduli" (classifying spaces).
And let's get "morphisms" out of the way, since the word appears even in swinnerton -dyer. In one sense it is a generic version of the several terms automorphism, isomorphism, homomorphism, homeomorphism, diffeomorphism,... i.e. a map that preserves some structure to be specified. In algebraic geometry it usually means a structure preserving map that is defined everywhere, as "holomorphic" is used in complex analysis to mean meromorphic and having no poles. so in swinnerton - dyer it probably means holomorphic map. but it is often defined where it is used.

Here is another reference:

Lectures on Riemann surfaces
proceedings of the College on Riemann Surfaces, International Centre for Theoretical Physics, Trieste, Italy, 9 Nov.-18 Dec., 1987
editors, M. Cornalba, X. Gomez-Mont, A. Verjovsky.
Published 1989 by World Scientific in Singapore, Teaneck, NJ .
Written in English.

Table of Contents

Complex analytic theory of Teichmüller space / R.M. Porter

Riemann surfaces, moduli, and hyperbolic geometry / S.A. Wolpert

Gauge theory on Riemann surfaces / N.J. Hitchin

Graph curves and curves on K3 surfaces / R. Miranda

Koszul cohomology and geometry / M.L. Green

Constructing the moduli space of stable curves / I. Morrison

Meromorphic functions and cohomology on a Riemann surface / X. Gomez-Mont

The theorems of Riemann-Roch and Abel / M. Cornalba

The Jacobian variety of a Riemann surface and its theta geometry / R. Smith

Families of varieties and the Hilbert scheme / C. Ciliberto and E. Sernesi

A sampling of vector bundle techniques in the study of linear series / R. Lazarsfeld

Moduli of curves and theta-characteristics / M. Cornalba

Some algebraic geometrical methods in string theory / L. Dabrowski and C. Reina

Lectures on stable curves / F. Bardelli.

Edition Notes
Includes bibliographical references.
"Revised versions of most of the original notes, covering the entire spectrum of the subjects touched upon during the College, from the foundational ones to areas of very active current research"--P. v.

Classifications
Library of Congress
QA333 .C65 1987