Siegel modular forms of small weight and the Witt operator

摘要

To calculate dimensions of Siegel modular forms including non-cusp forms, we determine the image of Siegel Phi-operator for small weight which were unknown in general theory. We treat the case of the Hecke type group of prime level and also vector valued Siegel modular forms of level one of degree two. For this purpose we propose a new basis problem on theta functions related with the Witt operator. We also show the surjectivity of the Witt operator in case of vector valued Siegel modular forms of level one for big weight by giving certain new dimension formulas of Siegel modular forms. We also give new upper and lower bounds of unknown dimensions of vector valued Siegel modular forms of small weight.