Congruences of Eisenstein series of level $\Gamma_1(N)$ via Dieudonn\'e theory of formal groups

Abstract

In this paper, we give a new explanation of congruences of Eisenstein series of level $\Gamma_1(N)$ and character $\chi$. 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 Dieudonn\'e theory of height $1$ formal $A$-modules and their finite subgroup schemes. We give a generalization of this Riemann-Hilbert correspondence in terms of formal groups of height greater than $1$.