Developments in Mathematics

Generalizations of Thomae's Formula for Zn Curves

Authors: Farkas, Hershel M., Zemel, Shaul

  • 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
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  • ISBN 978-1-4419-7846-2
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About this book

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.

Reviews

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)


Table of contents (6 chapters)

  • Riemann Surfaces

    Farkas, Hershel M. (et al.)

    Pages 1-30

  • Z n Curves

    Farkas, Hershel M. (et al.)

    Pages 31-74

  • Examples of Thomae Formulae

    Farkas, Hershel M. (et al.)

    Pages 75-141

  • Thomae Formulae for Nonsingular Z n Curves

    Farkas, Hershel M. (et al.)

    Pages 143-182

  • Thomae Formulae for Singular Z n Curves

    Farkas, Hershel M. (et al.)

    Pages 183-232

Buy this book

eBook $149.00
price for USA (gross)
  • ISBN 978-1-4419-7847-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $189.00
price for USA
  • ISBN 978-1-4419-7846-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $189.00
price for USA
  • ISBN 978-1-4614-2758-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Generalizations of Thomae's Formula for Zn Curves
Authors
Series Title
Developments in Mathematics
Series Volume
21
Copyright
2011
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-1-4419-7847-9
DOI
10.1007/978-1-4419-7847-9
Hardcover ISBN
978-1-4419-7846-2
Softcover ISBN
978-1-4614-2758-2
Series ISSN
1389-2177
Edition Number
1
Number of Pages
XVII, 354
Topics