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

Abstract

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group Γ 0 (p n ) with values in the second Milnor K-group (modulo 2-torsion) of the ring of p-integers of the cyclotomic extension Q(μ 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| 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.