Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
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 $ \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.


References [Enhancements On Off] (What's this?)

  • [AS86] Ash, A., Stevens, G.: Modular forms in characteristic $ \ell$ and special values of their $ L$-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 165), B. Mazur and G. Stevens, eds., 1991. MR 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 $ p$-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 $ p<1000$, http://abel.math.harvard.edu/$ \sim$sharifi/computations.html, http://math.arizona. edu/$ \sim$wmc 284.
  • [Mer94] Merel, L.: Universal Fourier expansions of modular forms, On Artin's conjecture for odd $ 2$-dimensional representations, Springer, Berlin, 1994, pp. 59-94. MR 1322319 (96h:11032)
  • [Mi71] Milnor, J.: Introduction to Algebraic $ K$-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., $ 4^{\hbox{e}}$ série 36 (2003), 225-269. MR 1980312 (2004d:11045)
  • [Sh1-04] Sharifi, R.: The various faces of a pairing on $ p$-units, slides from a talk at International Univ. Bremen on 5/10/04, http://www.math.mcmaster.ca/$ \sim$ sharifi/bremen.pdf.
  • [Sh2-04] Sharifi, R.: Computations on Milnor's $ K_2$ of Integer Rings, slides from a talk at Max Planck Institute of Mathematics on 5/17/04, http://www.math.mcmaster. ca/$ \sim$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 $ L$-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)

Similar Articles

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.

American Mathematical Society