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Mathematics - Algebra | Generalizations of Thomae's Formula for Zn Curves

Generalizations of Thomae's Formula for Zn Curves

Series: Developments in Mathematics, Vol. 21

Farkas, Hershel M., Zemel, Shaul

2011, XVII, 354 p.

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  • The first monograph to study generalizations of the Thomae Formulae to Zn curves
  • Provides an introduction to the basic principles of compact Riemann surfaces, theta functions, algebraic curves, and branch points
  • Examples support the theory and reveal the broad applicability of this theory to numerous other disciplines including conformal field theory, low dimensional topology, the theory of special functions

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.

Content Level » Research

Keywords » Algebraic Curves - Algebraic Geometry - Branch Points - Conformal Field Theory - Hypereliptic Curves - Riemann Surfaces - Theta Constants - Theta Functions - Thomae Formulae - Zn Curves

Related subjects » Algebra - Analysis - Number Theory and Discrete Mathematics

Table of contents 

- Introduction.- 1. Riemann Surfaces.- 2. Zn Curves.- 3. Examples of Thomae Formulae.- 4. Thomae Formulae for Nonsingular Zn Curves.- 5. Thomae Formulae for Singular Zn Curves.-6. Some More Singular Zn Curves.-Appendix A. Constructions and Generalizations for the Nonsingular and Singular Cases.-Appendix B. The Construction and Basepoint Change Formulae for the Symmetric Equation Case.-References.-List of Symbols.-Index.

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