On Dimension Formula for Siegel Modular Forms

Abstract

This chapter presents the dimension formula for Siegel modular forms. For general group Γ and representation μ, two main approaches are known. The first is a geometric one that uses Riemann–Roch–Hirzebruch's formula and the holomorphic Lefschetz fixed points formula, when Γ has fixed points. The second is a group-theoretical one that uses Selberg's trace formula. The chapter presents a problem to calculate the coefficients of the partial fractions of the generating function of A(Γ 3 (1)). There are two different standpoints to solve this problem. One is to calculate all the contributions of the conjugacy classes of Γ3 (1). The other is to calculate only contributions of conjugacy classes that are easy to calculate and determine the coefficients of some part of partial fractions, and to determine the coefficients of the remaining partial fractions by the knowledge of modular forms of lower weights instead of the calculation of the contributions of difficult conjugacy classes.