The Steinberg symbol and special values of L-functions

In this talk, we will describe a construction of Eisenstein classes in the parabolic cohomology of Γ₀(M) with values in Milnor's K₂-group of the ring of S-integers of the cyclotomic extension Q(μ_M). In the case where p is an irregular prime, we will explain a congruence between certain projections of Steinberg symbols of p-units and L-values of level-1 cusp forms congruent to an Eisenstein series mod p, a result that was predicted by a conjecture of R. Sharifi.