On a correspondence between p-adic Siegel-Eisenstein series and genus theta series
We prove that the Dirichlet series of Rankin–Selberg type associated with any pair of (not necessarily cuspidal) Siegel modular forms of degree and weight has meromorphic continuation to . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight may be expressed in terms of the residue at of the associated Dirichlet series.
Using the special values of Siegel modular functions, we construct Minkowski units for the ray class field of modulo . Our work is based on investigating the prime decomposition of the special values and describing explicitly the action of the Galois group for the special values. Futhermore we construct the full unit group of using modular and circular units under the GRH.
We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.