The Steinberg symbol and special values of $L$-functions

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group $\Gamma_0(p^n)$ with values in the second Milnor $K$-group (modulo $2$-torsion) of the ring of $p$-integers of the cyclotomic extension $\mathbb{Q}(\mu_{p^n})$. We endow this cohomology group with a natural action of the standard Hecke operators and discuss the existence of special Hecke eigenclasses in its parabolic cohomology. Moreover, for $n=1$, assuming the non-degeneracy of a certain pairing on $p$-units induced by the Steinberg symbol when $(p,k)$ is an irregular pair, i.e. $p\vert\frac{B_k}{k}$, we show that the values of the above pairing are congruent mod $p$ to the $L$-values of a weight $k$, level $1$ cusp form which satisfies Eisenstein-type congruences mod $p$, a result that was predicted by a conjecture of R. Sharifi.