  In this paper, we give a dimension formula for spaces of Siegel cusp forms of
general degree with respect to neat arithmetic subgroups. The formula was
conjectured before by several researchers. The dimensions are expressed by
special values of Shintani zeta functions for spaces of symmetric matrices at
non-positive integers. This formula was given by Shintani for only a small part
of the geometric side of the trace formula. To be precise, it is the
contribution of unipotent elements corresponding to the partitions
$(2^j,1^{2n-2j})$, where $n$ denotes the degree and $0\leq j \leq n$. Hence,
our work is to show that all the other contributions vanish. In addition, one
finds that Shintani's formula means the dimension itself. Combining our formula
and an explicit formula of the Shintani zeta functions, which was discovered by
Ibukiyama and Saito, we can derive an explicit dimension formula for the
principal congruence subgroups of level greater than two. In this explicit
dimension formula, the dimensions are described by degree $n$, weight $k$,
level $N$, and the Bernoulli numbers $B_m$.
