Application: Eta Quotients

In this chapter we study Dedekind’s eta function and its quotients. In Sect. 4.1 we introduce Dedekind’s eta function and give the basic properties and the modularity conditions of quotients of rescaled Dedekind’s eta functions. In Sect. 4.2, we use those properties to devise an algorithm that determines all eta quotients in a given modular form space. In Sect. 4.3 we give a formula for the Eisenstein parts of eta quotients that are modular forms. In Sect. 4.4, we use the results of Sect. 4.3 to determine eta quotients that can be expressed in terms of Eisenstein series. These results yield beautiful product to sum formulas, many of which were previously known. Additionally, if the Eisenstein part of an eta quotient is a single Eisenstein series, then a proper normalization yields a modular form with multiplicative coefficients. All eta quotients with this property were discovered by Martin in [42]. In Sect. 4.5, inspired by this we give some infinite families of eta quotients whose Eisenstein part can be given by a single Eisenstein series, and hence the proper normalization of the Eisenstein part is multiplicative. Section 4.6 is devoted to expressing newforms as linear combinations of eta quotients and Eisenstein series. Finally, in Sect. 4.7, we give bases of spaces of cusp forms of levels 12 and 16 in terms of eta quotients.