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Eta Products and Theta Series Identities

Günter Köhler (Autor)

Buch | Hardcover

XXII, 622 Seiten

2010 | 2011
Springer Berlin (Verlag)
978-3-642-16151-3 (ISBN)

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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 (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)

Erscheint lt. Verlag	8.12.2010
Reihe/Serie	Springer Monographs in Mathematics
Zusatzinfo	XXII, 622 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	1075 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	11-02, 11F20, 11F27, 11R11 • Eisenstein series • eta products • Hecke theta series • modular forms (one variable) • Number Theory • quadratic number fields • Sums of squares • Zahlentheorie
ISBN-10	3-642-16151-0 / 3642161510
ISBN-13	978-3-642-16151-3 / 9783642161513
Zustand	Neuware