The Steinberg symbol and special values of -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 with values in the second Milnor -group (modulo -torsion) of the ring of -integers of the cyclotomic extension . 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 , assuming the non-degeneracy of a certain pairing on -units induced by the Steinberg symbol when is an irregular pair, i.e. , we show that the values of the above pairing are congruent mod to the -values of a weight , level cusp form which satisfies Eisenstein-type congruences mod , 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.