Darstellung des Maaß-Raumes mit ultrasphärischen Differentialoperatoren

• Description of the Maass space by ultraspherical differential operators

Bleß, Stefan; Krieg, Aloys (Thesis advisor); Heim, Bernhard (Thesis advisor)

Aachen (2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

Maass introduced some subspace of the vector space of siegel modular forms in 1979 by demanding some special relation to the fourier-coefficients and Andrianov showed in the same year, that this subspace is invariant under hecke operators. He did this by complex calculations to the fourier-coefficients. Gallenkämper, Heim and Krieg found some alternative proof in 2016. They used differential operators and hecke operators concerning the jacobi group to the siegel modular group. First this thesis transfers known results to siegel modular forms to modular forms, for example to modular forms concerning the congruence subgroup $\Gamma_{2,0}(q)$, the paramodular group or the extensions of the paramodular group. Also multiplier systems are considered. Then some alternative description of the maass space, concerning the congruence subgroup $\Gamma_{2,0}(q)$ or the paramodular group, by differential operators is proved, which leads, based on the new results of Gallenkämper, Heim and Krieg in 2016, to the invariance of these maass spaces under hecke operators. These special differential operators are based on a generalization of the Gegenbauer polynomials and they map modular forms componentwise to modular forms. So it is possible to apply hecke operators on each component. Since the differential operators are based on degree-increasing polynomials, equating the componentwise image of a modular form and the uniqueness of a fourier expansion lead directly to the maass condition. This new description of the maass space is based on infinity many hecke operators, but this description also remains by omitting finite many operators. This result is proven in this work for the congruence subgroup $\Gamma_{2,0}(q)$ and the paramodular group with some restrictions and leads to the invariance of the related maass spaces under hecke operators.