This book provides a comprehensive overview of the theory of theta functions, as applied to compact Riemann surfaces, as well as the necessary background for understanding and proving the Thomae formulae.
The exposition examines the properties of a particular class of compact Riemann surfaces, i.e., the Zn curves, and thereafter focuses on how to prove the Thomae formulae, which give a relation between the algebraic parameters of the Zn curve, and the theta constants associated with the Zn curve.
Graduate students in mathematics will benefit from the classical material, connecting Riemann surfaces, algebraic curves, and theta functions, while young researchers, whose interests are related to complex analysis, algebraic geometry, and number theory, will find new rich areas to explore. Mathematical physicists and physicists with interests also in conformal field theory will surely appreciate the beauty of this subject.
From the reviews:
"This book provides a detailed exposition of Thomae's formula for cyclic covers of CP1, in the non-singular case and in the singular case for Zn curves of a particular shape. ? This book is written for graduate students as well as young researchers ? . In any case, the reader should be acquainted with complex analysis (in several variables), Riemann surfaces, and some elementary algebraic geometry. It is a very readable book. The theory is always illustrated with examples in a very generous mathematical style." (Juan M. Cerviño Mathematical Reviews, Issue 2012 f)
"In the book under review, the authors present the background necessary to understand and then prove Thomae's formula for Zn curves. ? The point of view of the book is to work out Thomae formulae for Zn curves from 'first principles', i.e. just using Riemann's theory of theta functions. ? the 'elementary' approach which is chosen in the book makes it a nice development of Riemann's ideas and accessible to graduate students and young researchers." (Christophe Ritzenthaler, Zentralblatt MATH, Vol. 1222, 2011)
