The Steinberg symbol and special values of 𝐿-functions

Busuioc, Cecilia

doi:10.1090/S0002-9947-08-04701-6

The Steinberg symbol and special values of $L$-functions
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by Cecilia Busuioc PDF

Trans. Amer. Math. Soc. 360 (2008), 5999-6015 Request permission

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.

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