Eta Products and Theta Series Identities

1. Front Matter

   Pages I-XXI

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2. Theoretical Background

   1. Front Matter

      Pages 1-1

      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

      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

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.

• 11-02, 11F20, 11F27, 11R11
• Eisenstein series
• Hecke theta series
• eta products
• modular forms (one variable)
• number theory
• quadratic number fields
• sums of squares

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)

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

  Günter Köhler

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