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We describe torsion classes in the first cohomology group of$\text{SL}_2(\mathbb{Z})$. In particular, we obtain generalized Dickson'sinvariants for p-power polynomial rings. Secondly, we describe torsion classesin the zero-th homology group of $\text{SL}_2(\mathbb{Z})$ as a module over thetorsion invariants. As application, we obtain various congruences betweencuspidal forms of level one and Eisenstein series.

Torsions in Cohomology of $\text{SL}_2(\mathbb{Z})$ and Congruence of Modular Forms