Congruences of Eisenstein series of level Γ 1 (N) via Dieudonné theory of formal groups

In this paper, we give a new explanation of congruences of Eisenstein series of level Γ 1 (N) and character χ. Our approach is based on Katz's algebro-geometric explanation of p-adic congruences of normalized Eisenstein series E 2k of level 1. One crucial step in our argument is to reformulate a Riemann–Hilbert correspondence in Katz's explanation in terms of Dieudonné theory of height 1 formal A-modules and their finite subgroup schemes. We further generalize this Riemann–Hilbert correspondence in terms of formal groups of height greater than 1.