Dimension formulas for spaces of vector-valued Siegel cusp forms of degree two

Abstract

We give a general arithmetic dimension formula for spaces of vector-valued Siegel cusp forms of degree two. Then, using this formula, we derive explicit dimension formulas for arithmetic subgroups of any level for each Q -form of Sp ( 2 ; R ) . Tsushima has already given the dimension formulas for some congruence subgroups of the split Q -form in Tsushima (1983, 1997) [32,33]. We obtain an alternative proof for his results by using the Selberg trace formula and the theory of prehomogeneous vector spaces. As for the non-split Q -forms, our results are new. We generalize the results and proofs given in Arakawa (1981) [1], Christian (1969, 1975, 1977) [5,6], Hashimoto (1983, 1984) [12,13], Morita (1974) [25] for the scalar-valued case to the vector-valued case using the Selberg trace formula.