Skip to main content
SpringerLink
Log in
Book cover

Eta Products and Theta Series Identities

  • Book
  • © 2011

Overview

Authors:
  1. Günter Köhler

    1. Institute of Mathematics, University of Würzburg, Würzburg, Germany

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • This monograph brings to the public the large number of identities which were found during the past 20 years
  • The majority of these identities is new and has not been published elsewhere
  • Presents more than hundred examples for the coincidence of theta series of weight 1 on three distinct quadratic fields
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (31 chapters)

  1. Front Matter

    Pages I-XXI
    Download chapter PDF

  2. Theoretical Background

    1. Front Matter

      Pages 1-1
      Download chapter PDF

  3. Theoretical background

    1. Dedekind’s Eta Function and Modular Forms

      • Günter Köhler
      Pages 3-30

    2. Eta Products

      • Günter Köhler
      Pages 31-37

    3. Eta Products and Lattice Points in Simplices

      • Günter Köhler
      Pages 39-54

    4. An Algorithm for Listing Lattice Points in a Simplex

      • Günter Köhler
      Pages 55-65

    5. Theta Series with Hecke Character

      • Günter Köhler
      Pages 67-79

    6. Groups of Coprime Residues in Quadratic Fields

      • Günter Köhler
      Pages 81-95

  4. Examples

    1. Front Matter

      Pages 97-97
      Download chapter PDF

    2. Ideal Numbers for Quadratic Fields

      • Günter Köhler
      Pages 99-112

    3. Eta Products of Weight \(\frac{1}{2}\) and \(\frac{3}{2}\)

      • Günter Köhler
      Pages 113-118

    4. Level 1: The Full Modular Group

      • Günter Köhler
      Pages 119-131

    5. The prime level N=2

      • Günter Köhler
      Pages 133-153

    6. The prime level N=3

      • Günter Köhler
      Pages 155-171

    7. Prime levels N=p≥5

      • Günter Köhler
      Pages 173-186

    8. Level N=4

      • Günter Köhler
      Pages 187-214

    9. Levels N=p 2 with Primes p≥3

      • Günter Köhler
      Pages 215-221

    10. Levels N=p 3 and p 4 for Primes p

      • Günter Köhler
      Pages 223-250

    11. Levels N=pq with Primes 3≤p<q

      • Günter Köhler
      Pages 251-265

    12. Weight 1 for Levels N=2p with Primes p≥5

      • Günter Köhler
      Pages 267-290
Back to top

Keywords

About this book

This monograph deals with products of Dedekind's eta function, with Hecke theta series on quadratic number fields, and with Eisenstein series. The author brings to the public the large number of identities that have been discovered over the past 20 years, the majority of which have not been published elsewhere. The book will be of interest to graduate students and scholars in the field of number theory and, in particular, modular forms. It is not an introductory text in this field. Nevertheless, some theoretical background material is presented that is important for understanding the examples in Part II of the book. In Part I relevant definitions and essential theorems -- such as a complete proof of the structure theorems for coprime residue class groups in quadratic number fields that are not easily accessible in the literature -- are provided. Another example is a thorough description of an algorithm for listing all eta products of given weight and level, together with proofs of some results on the bijection between these eta products and lattice simplices.

Reviews

From the reviews:

“This monograph serves as a leading reference on the theory of eta products and theta series identities. The systematic approach to the theory of modular forms in general and eta products in particular makes it a reader-friendly monograph for those who have basic knowledge about the theory.” (Wissam Raji, Mathematical Reviews, Issue 2012 a)

“In the book under review mainly a highly interesting special class of theta functions is investigated, the class of Hecke theta series for quadratic number fields. … Most of the identities in the later sections of this monograph are supposed to be new. Clearly this is a most valuable addition to the literature on modular forms, and the modular forms people must be most grateful to the author for his fine achievement.” (Jürgen Elstrodt, Zentralblatt MATH, Vol. 1222, 2011)

Authors and Affiliations

  • Institute of Mathematics, University of Würzburg, Würzburg, Germany

    Günter Köhler

Bibliographic Information

  • Book Title: Eta Products and Theta Series Identities

  • Authors: Günter Köhler

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-16152-0

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2011

  • Hardcover ISBN: 978-3-642-16151-3Published: 08 December 2010

  • Softcover ISBN: 978-3-642-26629-4Published: 27 January 2013

  • eBook ISBN: 978-3-642-16152-0Published: 15 January 2011

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XXII, 622

  • Topics: Number Theory, Algebraic Topology

Publish with us

Policies and ethics

Back to top

Search

Navigation