Modular forms arising from divisor functions
Klein forms are defined by means of Weierstrass \(\sigma\)-functions. They can be expressed by infinite products, and certain products of Klein forms are holomorphic modular forms on congruence subgroups of the modular group \(\text{SL}_2(\mathbb{Z})\) whose zeros are supported only at the cusps. The theory of basic hypergeometric series provides many identities between infinite \(q\)-products and infinite \(q\)-series whose coefficients are combinations of divisor sums of the type \[ E_r(n, m) = \sum_{{d\mid n}\atop{d\equiv r\pmod n}}^{} 1 - \sum_{{d\mid n}\atop {d\equiv - r \pmod n}}^{} 1. \] These identities can be restated in terms of Klein forms. In this way, the authors obtain holomorphic modular forms for many congruence groups whose Fourier coefficients are expressed by divisor sums. For several groups, these functions constitute bases of the spaces of holomorphic modular forms of weights 1 and 2. As an application, the representation numbers of integers by binary quadratic forms with discriminants \(-7\) and \(-11\) are expressed by means of divisor sums \(E_r(n, m)\). Reviewer's remark: In some cases the (combinations of) divisor sums \(E_r(n, m)\) are character sums, and the corresponding modular forms can be identified with Hecke theta series on certain imaginary quadratic fields.
scientific article; zbMATH DE number 5566254