Eta Products and Theta Series Identities, 2011 Springer Monographs in Mathematics Series

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.

Introduction.- Part I: Theoretical background.- 1. Dedekind’s eta function and modular forms.- 2. Eta products.- 3. Eta products and lattice points in simplices.- 4. An algorithm for listing lattice points in a simplex.- 5. Theta series with Hecke character.- 6. Groups of coprime residues in quadratic fields.- Part II: Examples.-7. Ideal numbers for quadratic fields.- 8 Eta products of weight .- 9. Level 1: The full modular group.- 10. The prime level N = 2.- 11. The prime level N = 3.- 12. Prime levels N = p ≥ 5.- 13. Level N = 4.- 14. Levels N = p2 with primes p ≥ 3.- 15 Levels N = p3 and p4 for primes p.- 16. Levels N = pq with primes 3 ≤ p < q.- 17. Weight 1 for levels N = 2p with primes p ≥ 5.- 18. Level N = 6.- 19. Weight 1 for prime power levels p5 and p6.- 20. Levels p2q for distinct primes p = 2 and q.- 21. Levels 4p for the primes p = 23 and 19.- 22. Levels 4p for p = 17 and 13.- 23. Levels 4p for p = 11 and 7.- 24. Weight 1 for level N = 20.- 25. Cuspidal eta products of weight 1 for level 12.- 26. Non-cuspidal eta products of weight 1 for level 12.- 27. Weight 1 for Fricke groups Γ∗(q3p).- 28. Weight 1 for Fricke groups Γ∗(2pq).- 29. Weight 1 for Fricke groups Γ∗(p2q2).- 30. Weight 1 for the Fricke groups Γ∗(60) and Γ∗(84).- 31. Some more levels 4pq with odd primes p _= q.- References.- Directory of Characters.- Index of Notations.- Index.

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