On some \(p\)-adic properties of Siegel-Eisenstein series. (Q1421302)

Nagaoka has shown in previous work that a certain \(p\)-adic Siegel-Eisenstein series of weight 1 and degree \(n\) (i.e., a formal power series arising as \(p\)-adic limit of Siegel-Eisenstein series of degree \(n\) and growing weight \(k_m = 1 + ((p-1)/2) p^{m-1}\) for the full modular group) has rational coefficients and is in fact a Siegel modular form of weight 1 and primitive nebentype for the group \(\Gamma_0^{(n)}(p)\). The authors prove here an analogous result for weight \(k > 1\) and degree \(n = 2\). The proof is given by calculating the Fourier coefficients and deriving an explicit formula that expresses the \(p\)-adic modular form that is a limit of degree 2 Eisenstein series of growing weight as a linear combination of Siegel-Eisenstein series of weight \(k\) and primitive nebentype for \(\Gamma_0^{(2)}(p)\). The main work goes into calculations of local densities.