The Steinberg Symbol and Special Values of L-Functions. / Busuioc, Cecilia.

In: Transactions of the American Mathematical Society, Vol. 360, No. 11, 11.2008, p. 5999–6015.

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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|\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.