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Tata lectures on theta. I. With the collaboration of C. Musili, M. Nori, E. Previato, and M. Stillman. Reprint of the 1983 edition. (English) Zbl 1112.14002
Modern Birkhäuser Classics. Basel: Birkhäuser (ISBN 978-0-8176-4572-4/pbk; 978-0-8176-4577-9/ebook). xiv, 236 p. (2007).
D. Mumford’s lectures on theta functions, once consecutively delivered at the Tata Institute of Fundamental Research (Mumbai, India), at Harvard University (Boston, USA), and at the University of Montreal (Canada) between 1978 and 1980, were published in two volumes by Birkhäuser Verlag in 1983 (Zbl 0509.14049) and 1984 (Zbl 0549.14014). Among many other research and survey monographs in mathematics issued by Birkhäuser, Mumford’s notes have become foundational standard texts in their field, so to speak modern classics. Through the new book series “Modern Birkhäuser Classics”, the publisher has begun to re-release a selected number of these modern classics in their original form, that is as entirely uncorrected reprints, in order to maintain their availability for further generations of mathematicians. According to their outstanding significance, D. Mumford’s famous “Tata Lectures on Theta I, II” have been chosen to be among the first volumes in this series of faithful reprints of modern classics, and the book under review is just Part I of Mumford’s standard work on theta functions.
The first edition of “Tata Lectures on Theta I” appeared in 1983 (Zbl 0509.14049) and was reviewed at length by S. Koizumi at that time. As the present edition is the unaltered reprint of the original, we may unrestrictedly refer to Koizumi’s review as for the finer details, and just recall that this volume comprises the first two chapters of Mumford’s lectures.
Chapter I serves as an introduction and motivation in that it discusses theta functions in one variable from the analytic viewpoint, whereas Chapter II provides the basic results on theta functions in several variables, including their geometric meaning with respect to Jacobians of compact Riemann surfaces and their relations to Siegel’s symplectic geometry.
Although, this sounds like the standard approach to the theory of theta functions, it ought to be emphasized again, in retrospective, that Mumford’s disposition is quite special and purposive. Namely, at that time when the study of the fascinating link between moduli of algebraic curves, theta functions, nonlinear partial differential equations of mathematical physics, Hamiltonian systems, and representations of Lie groups had reached its first peak, mainly driven by Krichever theory and new approaches to the long-standing Schottky problem, Mumford’s goal was to survey large parts of the classical theory of theta functions from this modern-viewpoint, but thereby starting from scratch and combining the classic analytic and the modern geometric aspects in a unifying way. He did so in his notorious masterly, enlightening style of teaching, which propelled the further developments in this field significantly, and that’s why his lectures on theta functions and their applications are so unique and of enduring topicality.

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14K25 Theta functions and abelian varieties
11F11 Holomorphic modular forms of integral weight
14H10 Families, moduli of curves (algebraic)
14H42 Theta functions and curves; Schottky problem