The dimensions of spaces of Siegel cusp forms of general degree

TL;DR

This paper derives an explicit dimension formula for spaces of Siegel cusp forms of general degree, expressing dimensions via special values of Shintani zeta functions, and confirms conjectured formulas for these spaces.

Contribution

It proves the full dimension formula for Siegel cusp forms by showing all other contributions vanish, extending previous partial results and providing explicit formulas involving Bernoulli numbers.

Findings

Dimension formula expressed by Shintani zeta functions

All contributions except unipotent elements vanish

Explicit formulas for principal congruence subgroups

Abstract

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 $B_m$.