The Steinberg symbol and special values of -functions

Author:
Cecilia Busuioc

Journal:
Trans. Amer. Math. Soc. **360** (2008), 5999-6015

MSC (2000):
Primary 11F67

DOI:
https://doi.org/10.1090/S0002-9947-08-04701-6

Published electronically:
June 26, 2008

MathSciNet review:
2425699

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[AS86]**Ash, A., Stevens, G.: Modular forms in characteristic and special values of their -functions,*Duke Mathematical Journal***53**, No. 3, 1986. MR**860675 (88h:11036)****[GS91]**Greenberg, R., Stevens, G.: On the conjecture of Mazur, Tate, and Teitelbaum, in*p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Contemporary Mathematics*, B. Mazur and G. Stevens, eds., 1991. MR**165)****1279598 (94m:11006)****[Ma72]**Manin, J.: Parabolic Points and Zeta Functions of Modular Curves,*Math. USSR Izvestija***36**, No. 1, 1972, 19-66. MR**0314846 (47:3396)****[Ma73]**Manin, J.: Periods of parabolic forms and -adic Hecke series,*Math. USSR Sbornik***21**, No. 3, 1973.**[MS03]**McCallum, W., Sharifi, R.: A Cup Product in Galois Cohomology,*Duke Mathematical Journal***120**, No. 2, 2003. MR**2019977 (2004j:11136)****[MS]**McCallum, W., Sharifi, R.:*Magma routines for computing the table of pairings for*, http://abel.math.harvard.edu/sharifi/computations.html, http://math.arizona. edu/wmc 284.**[Mer94]**Merel, L.: Universal Fourier expansions of modular forms,*On Artin's conjecture for odd -dimensional representations*, Springer, Berlin, 1994, pp. 59-94. MR**1322319 (96h:11032)****[Mi71]**Milnor, J.: Introduction to Algebraic -theory,*Annals of Mathematics Studies***72**, Princeton University Press, 1971. MR**0349811 (50:2304)****[Oh03]**Ohta, M.: Congruence modules related to Eisenstein series,*Ann. Scient. École Norm. Sup.*, série**36**(2003), 225-269. MR**1980312 (2004d:11045)****[Sh1-04]**Sharifi, R.: The various faces of a pairing on -units, slides from a talk at International Univ. Bremen on 5/10/04, http://www.math.mcmaster.ca/ sharifi/bremen.pdf.**[Sh2-04]**Sharifi, R.: Computations on Milnor's of Integer Rings, slides from a talk at Max Planck Institute of Mathematics on 5/17/04, http://www.math.mcmaster. ca/sharifi/dagslides.pdf.**[Sh3-05]**Sharifi, R.: Iwasawa Theory and the Eisenstein Ideal, Duke Math. J.**137**(2007), 63-101. MR**2309144****[Sh4-06]**Sharifi, R.: Cup Products and -values of Cusp Forms, preprint.**[St89]**Stevens, G.: The Eisenstein measure and real quadratic fields, in*The Proceedings of the International Number Theory Conference (Université Laval, 1987)*, J.-M. De Koninck and C. Levesque, eds., de Gruyter, 1989. MR**1024612 (90m:11077)****[W97]**Washington, L.C.: Introduction to Cyclotomic Fields,*Graduate Texts in Mathematics***83**, Springer, New York, 1997. MR**1421575 (97h:11130)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11F67

Retrieve articles in all journals with MSC (2000): 11F67

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.