On Siegel--Eisenstein series of level $p$ and their $p$-adic properties

We construct a Siegel–Eisenstein series of level $p$ with a quadratic character mod $p$ which is a $U(p)$-eigenfunction with eigenvalue $1$, and calculate its Fourier coefficients explicitly. We show that this Siegel–Eisenstein series is a $p$-adic Siegel–Eisenstein series, i.e., it is a $p$-adic limit of a sequence of Siegel–Eisenstein series of level $1$. We prove also that the Siegel–Eisenstein series with a nonquadratic character mod $p$ constructed by Takemori is also a $p$-adic Siegel–Eisenstein series.