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

TL;DR

This paper offers a new algebraic geometric perspective on congruences of Eisenstein series of level mma_1(N) by connecting Katz's p-adic methods with Dieudonne9 theory of formal groups, generalizing existing correspondences.

Contribution

It introduces a novel reformulation of the Riemann-Hilbert correspondence using Dieudonne9 theory for formal groups of arbitrary height, extending Katz's approach.

Findings

Provides a new explanation for Eisenstein series congruences

Generalizes Riemann-Hilbert correspondence to higher height formal groups

Connects formal group theory with modular form congruences

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 $1$.