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Transactions of the American Mathematical Society

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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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  • [AS86] Avner Ash and Glenn Stevens, Modular forms in characteristic 𝑙 and special values of their 𝐿-functions, Duke Math. J. 53 (1986), no. 3, 849–868. MR 860675, 10.1215/S0012-7094-86-05346-9
  • [GS91] Barry Mazur and Glenn Stevens (eds.), 𝑝-adic monodromy and the Birch and Swinnerton-Dyer conjecture, Contemporary Mathematics, vol. 165, American Mathematical Society, Providence, RI, 1994. Papers from the workshop held at Boston University, Boston, Massachusetts, August 12–16, 1991. MR 1279598
  • [Ma72] Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
  • [Ma73] Manin, J.: Periods of parabolic forms and $ p$-adic Hecke series, Math. USSR Sbornik 21, No. 3, 1973.
  • [MS03] William G. McCallum and Romyar T. Sharifi, A cup product in the Galois cohomology of number fields, Duke Math. J. 120 (2003), no. 2, 269–310. MR 2019977, 10.1215/S0012-7094-03-12023-2
  • [MS] McCallum, W., Sharifi, R.: Magma routines for computing the table of pairings for $ p<1000$,$ \sim$sharifi/computations.html, http://math.arizona. edu/$ \sim$wmc 284.
  • [Mer94] Loïc Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, pp. 59–94. MR 1322319, 10.1007/BFb0074110
  • [Mi71] John Milnor, Introduction to algebraic 𝐾-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. MR 0349811
  • [Oh03] Masami Ohta, Congruence modules related to Eisenstein series, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 225–269 (English, with English and French summaries). MR 1980312, 10.1016/S0012-9593(03)00009-0
  • [Sh1-04] Sharifi, R.: The various faces of a pairing on $ p$-units, slides from a talk at International Univ. Bremen on 5/10/04,$ \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] Romyar T. Sharifi, Iwasawa theory and the Eisenstein ideal, Duke Math. J. 137 (2007), no. 1, 63–101. MR 2309144, 10.1215/S0012-7094-07-13713-X
  • [Sh4-06] Sharifi, R.: Cup Products and $ L$-values of Cusp Forms, preprint.
  • [St89] Glenn Stevens, The Eisenstein measure and real quadratic fields, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 887–927. MR 1024612
  • [W97] Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575

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Additional Information

Cecilia Busuioc
Affiliation: Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 02215

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.