# Even Galois representations and the cohomology of GL(2,Z)

## TL;DR

This paper constructs a connection between certain even Galois representations induced from characters of real quadratic fields and Hecke eigenclasses in the cohomology of GL(2,Z), revealing new links in number theory.

## Contribution

It introduces a method to attach specific even Galois representations to cohomology classes of GL(2,Z) under new conditions on the inducing characters.

## Key findings

- Established a correspondence between induced Galois representations and cohomology classes.
- Extended the understanding of the relationship between Galois representations and automorphic forms.
- Provided new examples of Galois representations linked to cohomological data.

## Abstract

Let $\rho$ be a two-dimensional even Galois representation which is induced from a character $\chi$ of odd order of the absolute Galois group of a real quadratic field. After imposing some additional conditions on $\chi$, we attach $\rho$ to a Hecke eigenclass in the cohomology of ${\rm GL}(2,\mathbb Z)$ with coefficients in a certain infinite-dimensional vector space over a field of characteristic not equal to 2.