2 Siegel modular forms and the holomorphic projection operator

Abstract

2.1 Siegel modular forms and Hecke operators 2.1.1 Symplectic group and Siegel upper half plane 2.1.2 Siegel modular forms 2.1.3 The Hecke algebra 2.1.4 Hecke operators 2.1.5 Hecke polynomials 2.1.6 The spinor zeta function and the standard zeta function 2.1.7 Non-commutative extension of the Hecke algebra and the Satake isomorphism 2.1.8 Action of the Hecke operators on Fourier expansions 2.2 Theta series, Siegel-Eisenstein series and the Rankin zeta function 2.2.1 Theta series 2.2.2 Siegel-Eisenstein series 2.2.3 The Rankin zeta function 2.2.4 The standard zeta function D(s,f,χ ) as the Rankin convolution 2.3 Formulas for Fourier coefficients of the Siegel-Eisenstein series 2.3.1 Rationality properties of Fourier coefficients 2.3.2 Preparation: the confluent hypergeometric function 2.3.3 Critical values of the confluent hypergeometric function 2.3.4 Normalized Siegel-Eisenstein series 2.4 Holomorphic projection and Maass operator 2.4.1 Holomorphic projection operator 2.4.2 Poincare series of two variables (of exponential type) of higher level 2.4.3 Reduction of theorem 2.16 to properties of Poincare series 2.4.4 Fourier expansion of the holomorphic projection of special modular forms 2.5 Explicit description of differential operators 2.5.1 The polynomial R(z;r,β )