The Steinberg symbol and special values of $L$-functions
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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.References
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Additional Information
- Cecilia Busuioc
- Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
- Email: celiab@math.bu.edu
- Received by editor(s): October 27, 2006
- Published electronically: June 26, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 5999-6015
- MSC (2000): Primary 11F67
- DOI: https://doi.org/10.1090/S0002-9947-08-04701-6
- MathSciNet review: 2425699