Siegel modular forms and the holomorphic projection operator

Abstract

This chapter contains some preparatory facts which we shall use in the construction of non-Archimedean standard zeta-functions in the next chapter. We recall main properties of Siegel modular forms and of the action of the Hecke algebra on them, as well as the definitions of spinor zeta functions and standard zeta functions (§1), see also [An2], [An7]. Then in §2 we present some standard results on theta series with a Dirichlet character [An-M1], [An-M2], [St2] and recall the definitions of Siegel — Eisenstein series, and of Rankin type convolutions of Siegel modular forms. Also, we recall their relation with the standard zeta functions (Andrianov’s identity). In §3 we give an exposition of some recent results of Shimura and P.Feit on real analytic Siegel — Eisenstein series and their analytic continuation in terms of confluent hypergeometric functions [Fe], [Shi7], [Shi9]. These results extend previous results of V.L.Kalinin [K1] and Langlands [Ll1]. In the final §4 a detailed study of holomorphic projection operator and its basic properties is given. The formula of theorem 4.2 provides an explicit formula for computing the holomorphic projection onto the space of holomorphic (not necessarily cusp) modular forms for functions belonging to a wide class of C ^∞-Siegel modular forms. Ealier the holomorphic projection operator onto the space of cusp form was studied by J.Sturm [St1], [St2], B.Gross and D.Zagier [Gr-Z] under some restrictive assumptions on the growth of modular forms. Theorem 4.6 gives an explicit description of the action of this operator in terms of Fourier expansions. Here we also establish a very explicit formula (3.36) for the special (critical) values of the confluent hypergeometric function.