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In this paper, we first explain congruences of Eisenstein series of level$\Gamma_1(N)$ and character $\chi$. Our approach is based on Katz'salgebro-geometric explanation of $p$-adic congruences of normalized Eisensteinseries $E_{2k}$ of level $1$. One crucial step in our argument is toreformulate a Riemann-Hilbert correspondence in Katz's explanation in terms ofDieudonn\'e theory of height $1$ formal $A$-modules and their finite subgroupschemes.  We further connect congruences of modular forms in the Eisenstein subspace$\mathcal{E}_{k}(\Gamma_1(N),\chi)$ with certain group cohomology involving theDirichlet character $\chi$. When $\chi$ is trivial, this group cohomologycomputes the image of the $J$-homomorphism in the stable homotopy groups ofspheres. We have therefore connected congruences of Eisenstein series $E_{2k}$of level $1$ to the image of $J$.

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