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Transactions of the American Mathematical Society

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Generalized reciprocity laws

Authors: José M. Muñoz Porras and Fernando Pablos Romo
Journal: Trans. Amer. Math. Soc. 360 (2008), 3473-3492
MSC (2000): Primary 14H05, 19F15, 14M15
Published electronically: February 27, 2008
MathSciNet review: 2386233
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Abstract: The aim of this paper is to give an abstract formulation of the classical reciprocity laws for function fields that could be generalized to the case of arbitrary (non-commutative) reductive groups as a first step to finding explicit non-commutative reciprocity laws. The main tool in this paper is the theory of determinant bundles over adelic Sato Grassmannians and the existence of a Krichever map for rank $ n$ vector bundles.

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  • 1. Álvarez, A., Drinfeld Moduli Schemes and Infinite Grassmannians, J. Algebra 225, (2000) 822-835. MR 1741564 (2001e:11059a)
  • 2. Álvarez Vázquez, A.; Muñoz Porras, J. M.; Plaza Martın, F. J., The Algebraic Formalism of Soliton Equations over Arbitrary Base Fields, Aportaciones Matemáticas: Taller de Variedades Abelianas y Funciones Theta; Sociedad Matemática Mexicana 13, (1998) 3-40. MR 1781698 (2002d:14051)
  • 3. Anderson, G. W.; Pablos Romo, F., Simple Proofs of Classical Explicit Reciprocity Laws on Curves using Determinant Groupoids over an Artinian Local Ring, Comm. Algebra 32(1), (2004) 79-102. MR 2036223 (2005d:11099)
  • 4. Arbarello, E.; de Concini, C.; Kac, V.G., The Infinite Wedge Representation and the Reciprocity Law for Algebraic Curves, Proc. of Symposia in Pure Mathematics, Volume 49, Part I, A.M.S., (1989) 171-190. MR 1013132 (90i:22034)
  • 5. Beilinson, A.; Bloch S.; Esnault H., $ \epsilon$-factors for Gauss-Manin determinants, Moscow Math. J. 2(3), (2002) 477-532. MR 1988970 (2004m:14011)
  • 6. Beilinson, A.; Drinfeld, V., Quantization of Hitchin's Integrable System and Hecke Eigensheaves, manuscript available at arinkin/langlands/.
  • 7. Contou-Carrère, C., Jacobienne Locale, Groupe de Bivecteurs de Witt Universel, et Symbole Modéré, C.R. Acad. Sci. Paris, t. 318, Série I (1994) 743-746. MR 1272340 (95c:14059)
  • 8. Kapranov, M., Semiinfinite symmetric powers, e-print: math.QA/0107089.
  • 9. Knudsen, F.; Mumford, D.,The projectivity of the moduli space of stable curves I: Preliminaries on det and Div., Math. Scand. 39, (1976) 19-55. MR 0437541 (55:10465)
  • 10. Pablos Romo, F., A Generalization of the Contou-Carrère Symbol, Israel J. Math. 141, (2004) 39-60. MR 2063024 (2005g:11115)
  • 11. Pablos Romo, F., On the Tame Symbol of an Algebraic Curve, Comm. Algebra 30(9), (2002) 4349-4368. MR 1936475 (2003k:14042)
  • 12. Pressley, A.; Segal, G., Loop Groups, Oxford Univ. Press, Oxford (1986). MR 900587 (88i:22049)
  • 13. Segal, G.; Wilson, G., Loop groups and equations of KdV type, I.H.E.S., Publications Mathématiques 61, (1985) 5-64. MR 783348 (87b:58039)
  • 14. Tate, J., Residues of Differentials on Curves, Ann. Scient. École. Norm. Sup., 4a série 1, (1968) 149-159. MR 0227171 (37:2756)
  • 15. Weil, A., Généralisation des fonctions abéliennes, J. Math. Pures et Appl. 17, (1938) 47-87.
  • 16. Witten, E., Quantum Field Theory, Grassmannians and Algebraic Curves, Comm. Math. Phys. 113, (1988) 529-600. MR 923632 (88m:81127)

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Additional Information

José M. Muñoz Porras
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, Salamanca 37008, Spain

Fernando Pablos Romo
Affiliation: Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, Salamanca 37008, Spain

Received by editor(s): March 8, 2006
Published electronically: February 27, 2008
Additional Notes: This work was partially supported by DGI research contract no. MTM2006-07618 and Castilla y León regional government contract SA071/04.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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