## Eta Products and Theta Series Identities by Günter Köhler

By Günter Köhler

This monograph offers with items of Dedekind's eta functionality, with Hecke theta sequence on quadratic quantity fields, and with Eisenstein sequence. the writer brings to the general public the massive variety of identities which were found during the last twenty years, nearly all of that have no longer been released in other places. The publication may be of curiosity to graduate scholars and students within the box of quantity idea and, particularly, modular kinds. it's not an introductory textual content during this box. however, a few theoretical historical past fabric is gifted that's vital for realizing the examples partially II. partly I suitable definitions and crucial theorems -- corresponding to a whole evidence of the constitution theorems for coprime residue category teams in quadratic quantity fields that aren't simply obtainable within the literature -- are supplied. one other instance is a radical description of an set of rules for directory all eta items of given weight and point, including proofs of a few effects at the bijection among those eta items and lattice simplices.

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**Example text**

9. A version of our proof was given in [120]; here we will choose another approach. The exponents am in an eta product of level N form vectors in a space of dimension τ (N ). 9) from the cones K(ps ) and simplices S(ps , k) for the prime powers ps dividing N . Therefore we begin with studying prime power levels. The compactness of K(ps ) will follow from an explicit description of the edges of K(ps ). We shall see that the cones and simplices grow larger when the primes get smaller (while the exponent s is ﬁxed).

0 0 0 0 ... ... ... ... ...

An interesting class consists of those meromorphic modular forms whose poles are supported by the cusps, that is, which are holomorphic on H. Eta products belong to this class. The case of weight k = 0 is of foremost importance. A meromorphic modular form f of weight 0 and trivial multiplier system 1 for Γ is called a modular function for Γ. It satisﬁes f (Lz) = f (z) for all L ∈ Γ. Clearly, the set of all modular functions for Γ is a ﬁeld. It can be identiﬁed with the ﬁeld of meromorphic functions on the compact Riemann surface corresponding to Γ.