Steven G. Krantz
David Saltman (Chair)
2000 Mathematics Subject Classification. Primary 30F35, 30F30, 11F20, 11F25, 11F30,
11P81, 11P82, 11P83, 14H42, 14H45, 14H55, 20H10.
ABSTRACT. The book is an introduction to the theory of Theta Functions and Theta Constants
with rational characteristics showing their close connections to Riemann Surfaces and the Modular
Group. Applications to uniformization theorems, partition identities and combinatorial number
theory are developed. Projective embeddings of modular curves and Ramanujan partition congru-
ences are explored. Theta identities and their combinatorial contents are studied. Many examples
are provided, and directions for future studies are indicated.
Library of Congress Cataloging-in-Publication Dat a
Farkas, Hershel M.
Theta constants, Riemann surfaces, and the modular group : an introduction with applications
to uniformization theorems, partition identities and combinatorial number theory / Hershel M.
Farkas, Irwin Kra.
p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 37)
Includes bibliographical references and index.
ISBN 0-8218-1392-7 (acid-free paper)
1. Functions, Theta. 2. Riemann surfaces. 3. Modular groups. I. Kra, Irwin. II. Title.
QA345 .F37 2001
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in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Assistant to the Publisher, American Mathematical Society,
P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to
© 2001 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
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