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The Steinberg Symbol and Special Values of L-functions

Cecilia Busuioc
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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. $p|\frac{B_k}{k}$, 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.

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Cecilia Busuioc. 2006-10-27. The Steinberg Symbol and Special Values of L-functions. https://arxiv.org/abs/math/0609588

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