Eta Products and Theta Series Identities (Springer Monographs in Mathematics)

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. 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. 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. Günter Köhler. Includes Bibliographical References (p. 611-618) And Index. Front Matter....Pages I-XXI Front Matter....Pages 1-1 Dedekind’s Eta Function and Modular Forms....Pages 3-30 Eta Products....Pages 31-37 Eta Products and Lattice Points in Simplices....Pages 39-54 An Algorithm for Listing Lattice Points in a Simplex....Pages 55-65 Theta Series with Hecke Character....Pages 67-79 Groups of Coprime Residues in Quadratic Fields....Pages 81-95 Front Matter....Pages 97-97 Ideal Numbers for Quadratic Fields....Pages 99-112 Eta Products of Weight $\frac{1}{2}$ and $\frac{3}{2}$ ....Pages 113-118 Level 1: The Full Modular Group....Pages 119-131 The prime level N =2....Pages 133-153 The prime level N =3....Pages 155-171 Prime levels N = p ≥5....Pages 173-186 Level N =4....Pages 187-214 Levels N = p 2 with Primes p ≥3....Pages 215-221 Levels N = p 3 and p 4 for Primes p ....Pages 223-250 Levels N = pq with Primes 3≤ p < q ....Pages 251-265 Weight 1 for Levels N =2 p with Primes p ≥5....Pages 267-290 Level N =6....Pages 291-304 Weight 1 for Prime Power Levels p 5 and p 6 ....Pages 305-318 Levels p 2 q for distinct primes p ≠2 and q ....Pages 319-346 Front Matter....Pages 97-97 Levels 4 p for the Primes p =23 and 19....Pages 347-368 Levels 4 p for p =17 and 13....Pages 369-396 Levels 4 p for p =11 and 7....Pages 397-425 Weight 1 for level N =20....Pages 427-454 Cuspidal Eta Products of Weight 1 for Level 12....Pages 455-484 Non-cuspidal Eta Products of Weight 1 for Level 12....Pages 485-512 Weight 1 for Fricke Groups Γ ∗ ( q 3 p )....Pages 513-525 Weight 1 for Fricke Groups Γ ∗ (2 pq )....Pages 527-545 Weight 1 for Fricke Groups Γ ∗ ( p 2 q 2 )....Pages 547-558 Weight 1 for the Fricke Groups Γ ∗ (60) and Γ ∗ (84)....Pages 559-570 Some More Levels 4 pq with Odd Primes p ≠ q ....Pages 571-591 Back Matter....Pages 593-621