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

Abstract

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