Dimension Formulas of Siegel Modular Forms of Weight 3 and Supersingular Abelian Surfaces Revised version
In this article, first we give dimension formulas of the space S3(G) of Siegel cusp forms of weight 3 of degree 2 belonging to discrete subgroups G of parahoric type, including Hecke type groups G0(p) paramodular groups K(p) Klingen type groups G 0(p) and the Iwahori subgroup B(p) of any prime level. As for weight k 4, the dimension formulas for Siegel modular forms of degree two are explicitly known for many discrete subgroups and there are two well known methods to calculate these: the Riemann Roch Hirzebruch theorem for k 4 and the Selberg trace formula for k 5. Both methods does not work for weight k 3 in general and there are no general ways to calculate the dimension for k 3. Before our present work, the dimensions of S3(G) were calculated only for finitely many (conjugacy classes of) discrete subgroups by rather complicated technical numerical calculation depending heavily on each discrete group G (cf. e.g. Poor and Yuen [23] In this paper, we use holomorphic Lefschetz formula (a group invariant version of RiemannRoch Hirzebruch) The obstruction of the cohomology in this formula does not vanish in general, but by showing that it vanishes for the above discrete groups, we obtain our dimension formula. As for paramodular groups and Hecke type groups, our formula coincides with those conjectured in [17] and [7] Secondly, we give a geometric meaning of the dimensions of weight 3. We shall show that the dimensions of cusp forms of weight 3 is related with the geometry of principally polarized super singular abelian surfaces in several ways. It was known that certain arithmetic invariants of the locus of supersingular abelian surfaces in the moduli of principally polarized abelian surfaces over algebraically closed field of characteristic p are related with some class numbers of the compact real form of the split symplectic group Sp(2,R) of size four(due to Katsura, Oort and partly myself) These class numbers