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- W2004305763 abstract "The main results of this article concern the definition of a compactly 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." @default.
- W2004305763 created "2016-06-24" @default.
- W2004305763 creator A5022557244 @default.
- W2004305763 date "2008-06-26" @default.
- W2004305763 modified "2023-09-30" @default.
- W2004305763 title "The Steinberg symbol and special values of $L$-functions" @default.
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- W2004305763 doi "https://doi.org/10.1090/s0002-9947-08-04701-6" @default.
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