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The Steinberg symbol and special values of $ L$-functions


Author: Cecilia Busuioc
Journal: Trans. Amer. Math. Soc. 360 (2008), 5999-6015
MSC (2000): Primary 11F67
Published electronically: June 26, 2008
MathSciNet review: 2425699
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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\vert\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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Additional Information

Cecilia Busuioc
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215
Email: celiab@math.bu.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04701-6
Received by editor(s): October 27, 2006
Published electronically: June 26, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.