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.
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.
Library of Congress
QA333 .C65 1987