Eta Products and Theta Series Identities

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 (q3p).- 28. Weight 1 for Fricke groups (2pq).- 29. Weight 1 for Fricke groups (p2q2).- 30. Weight 1 for the Fricke groups (60) and (84).- 31. Some more levels 4pq with odd primes p _= q.- References.- Directory of Characters.- Index of Notations.- Index.

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)

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)