On the congruences of Eisenstein series with polynomial indexes

In this paper, based on Serre’s $p$-adic family of Eisenstein series, we prove a general family of congruences for Eisenstein series $G_k$ in the form $$\sum _{i=1}^n{g}_i(p)G_{f_i(p)}\equiv g_0(p)\mathrm{m}\mathrm{o}\mathrm{d}{p}^N,$$ where $f_1(t),\dots ,f_n(t)\in \mathbb{Z}[t]$ are non-constant integer polynomials with positive leading coefficients and $g_0(t),\dots ,g_n(t)\in \mathbb{Q}(t)$ are rational functions. This generalizes the classical von Staudt-Clausen’s and Kummer’s congruences of Eisenstein series, and also yields some new congruences.