The Steinberg symbol and special values of functions
Author:
Cecilia Busuioc
Journal:
Trans. Amer. Math. Soc. 360 (2008), 59996015
MSC (2000):
Primary 11F67
Published electronically:
June 26, 2008
MathSciNet review:
2425699
Fulltext 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 nondegeneracy 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 Eisensteintype congruences mod , a result that was predicted by a conjecture of R. Sharifi.
 [AS86]
Avner
Ash and Glenn
Stevens, Modular forms in characteristic 𝑙 and special
values of their 𝐿functions, Duke Math. J. 53
(1986), no. 3, 849–868. MR 860675
(88h:11036), http://dx.doi.org/10.1215/S0012709486053469
 [GS91]
Barry
Mazur and Glenn
Stevens (eds.), 𝑝adic monodromy and the Birch and
SwinnertonDyer 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
(94m:11006)
 [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
(47 #3396)
 [Ma73]
Manin, J.: Periods of parabolic forms and 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
(2004j:11136), http://dx.doi.org/10.1215/S0012709403120232
 [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]
Loïc
Merel, Universal Fourier expansions of modular forms, On
Artin’s conjecture for odd 2dimensional representations, Lecture
Notes in Math., vol. 1585, Springer, Berlin, 1994,
pp. 59–94. MR 1322319
(96h:11032), http://dx.doi.org/10.1007/BFb0074110
 [Mi71]
John
Milnor, Introduction to algebraic 𝐾theory, Princeton
University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.
Annals of Mathematics Studies, No. 72. MR 0349811
(50 #2304)
 [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
(2004d:11045), http://dx.doi.org/10.1016/S00129593(03)000090
 [Sh104]
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.
 [Sh204]
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.
 [Sh305]
Romyar
T. Sharifi, Iwasawa theory and the Eisenstein ideal, Duke
Math. J. 137 (2007), no. 1, 63–101. MR 2309144
(2008e:11135), http://dx.doi.org/10.1215/S001270940713713X
 [Sh406]
Sharifi, R.: Cup Products and 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
(90m:11077)
 [W97]
Lawrence
C. Washington, Introduction to cyclotomic fields, 2nd ed.,
Graduate Texts in Mathematics, vol. 83, SpringerVerlag, New York,
1997. MR
1421575 (97h:11130)
 [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 padic Monodromy and the Birch and SwinnertonDyer 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, 1966. 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. 5994. 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), 225269. MR 1980312 (2004d:11045)
 [Sh104]
 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.
 [Sh204]
 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.
 [Sh305]
 Sharifi, R.: Iwasawa Theory and the Eisenstein Ideal, Duke Math. J. 137 (2007), 63101. MR 2309144
 [Sh406]
 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)
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:
http://dx.doi.org/10.1090/S0002994708047016
PII:
S 00029947(08)047016
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.
