The Steinberg symbol and special values of 𝐿-functions

Source abstract

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group Γ 0 ( p n ) \Gamma _0(p^n) with values in the second Milnor K K -group (modulo 2 2 -torsion) of the ring of p p -integers of the cyclotomic extension Q ( μ p n ) \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 n=1 , assuming the non-degeneracy of a certain pairing on p p -units induced by the Steinberg symbol when ( p , k ) (p,k) is an irregular pair, i.e. p | B k k p|\frac {B_k}{k} , we show that the values of the above pairing are congruent mod p p to the L L -values of a weight k k , level 1 1 cusp form which satisfies Eisenstein-type congruences mod p p , a result that was predicted by a conjecture of R. Sharifi.