Galois representations and Hecke operators associated with the mod $p$ cohomology of $GL(1,\mathbb {Z})$ and $GL(2,\mathbb {Z})$

We prove that any Hecke eigenclass in the mod $p$ cohomology of a congruence subgroup of $GL(1,\mathbb {Z})$ or $GL(2,\mathbb {Z})$ has attached to it a mod $p$ Galois representation such that the characteristic polynomial of a Frobenius element at a prime $l$ equals the Hecke polynomial at $l$.