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

TL;DR

This paper investigates torsion classes in the cohomology of SL_2(Z), deriving new invariants and exploring their implications for congruences between modular forms.

Contribution

It introduces generalized Dickson's invariants for p-power polynomial rings and analyzes torsion classes in cohomology and homology of SL_2(Z).

Findings

Derived generalized Dickson's invariants for p-power polynomial rings.

Described torsion classes in cohomology and homology of SL_2(Z).

Established congruences between level one cuspidal forms and Eisenstein series.

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.