Generalizations Of Thomae's Formula For Zn [z Subscript N] Curves Generalizations Of Thomae's Formula For Zn Cuves

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. Previous publications on the generalization of the Thomae formulae to Zn curves have emphasized the theory's implications in mathematical physics and depended heavily on applied mathematical techniques. This book redevelops these previous results demonstrating how they can be derived directly from the basic properties of theta functions as functions on compact Riemann surfaces. "Generalizations of Thomae's Formula for Zn Curves" includes several refocused proofs developed in a generalized context that is more accessible to researchers in related mathematical fields such as algebraic geometry, complex analysis, and number theory. This book is intended for mathematicians with an interest in complex analysis, algebraic geometry or number theory as well as physicists studying conformal field theory. Front Matter....Pages i-xvii Riemann Surfaces....Pages 1-30 Z n Curves....Pages 31-74 Examples of Thomae Formulae....Pages 75-141 Thomae Formulae for Nonsingular Z n Curves....Pages 143-182 Thomae Formulae for Singular Z n Curves....Pages 183-232 Some More Singular Z n Curves....Pages 233-288 Back Matter....Pages 289-354 Riemann Surfaces -- Zn Curves -- Examples Of Thomae Formulae -- Thomae Formulae For Nonsingular Zn Curves -- Thomae Formulae For Singular Zn Curves -- Some More Singular Zn Curves. Hershel M. Farkas, Shaul Zemel. Includes Bibliographical References (p. 347) And Index.