We investigate the notion of equivariant forms as functions on the upper half-plane commuting with the action of a discrete group. We put an emphasis on the rational equivariant forms for a modular subgroup that are parametrized by generalized modular forms. Furthermore, we study this parametrization when the modular subgroup is of genus zero as well as their behavior under the effect of the Schwarz derivative.
Abstract. In this paper we investigate the zeros of the Eisenstein series E 2 . In particular, we prove that E 2 has infinitely many SL 2 (Z)-inequivalent zeros in the upper half-plane H, yet none in the standard fundamental F. Furthermore, we go on to investigate other fundamental regions in the upper half-plane H for which there do or do not exist zeros of E 2 . We establish infinitely many such regions containing a zero of E 2 and infinitely many which do not.
We describe the Schwarzian equations for the 328 completely replicable functions with integral q-coefficients [Ford et al., 'More on replicable functions ', Comm. Algebra 22 (1994) no. 13, 5175-5193].
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