The two-dimensional Contou-Carrère symbol and reciprocity laws

We define a two-dimensional Contou-Carrere symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrere symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We also relate the two-dimensional Contou-Carrere symbol to the two-dimensional class field theory.

[1]  Zhen-bing Luo,et al.  On the Contou-Carrere Symbol for Surfaces , 2013, 1310.7065.

[2]  C. Contou-Carrère.  Jacobienne locale d'une courbe formelle relative , 2013 .

[3]  D. Osipov,et al.  A categorical proof of the Parshin reciprocity laws on algebraic surfaces , 2010, 1002.4848.

[4]  Kazuya Kato,et al.  Modulus of a rational map into a commutative algebraic group , 2010, 1003.3289.

[5]  Ambrus Pál.  On the Kernel and the Image of the Rigid Analytic Regulator in Positive Characteristic , 2009, 0912.3115.

[6]  E. Frenkel,et al.  Gerbal Representations of Double Loop Groups , 2008, 0810.1487.

[7]  Lawrence Breen.  Bitorseurs et Cohomologie Non Abélienne , 2007 .

[8]  V. Drinfeld.  Infinite-Dimensional Vector Bundles in Algebraic Geometry , 2006 .

[9]  M. Kapranov,et al.  Formal loops II: A local Riemann–Roch theorem for determinantal gerbes , 2005, math/0509646.

[10]  D. Osipov.  Adeles on n-dimensional schemes and categories C_n , 2005, math/0509189.

[11]  D. Osipov.  Central extensions and reciprocity laws on algebraic surfaces , 2005, math/0501155.

[12]  V. Drinfeld.  Infinite-dimensional vector bundles in algebraic geometry (an introduction) , 2003, math/0309155.

[13]  G. Anderson,et al.  Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids Over an Artinian Local Ring , 2002, math/0207311.

[14]  A. Beilinson,et al.  $ε$-factors for Gauss-Manin determinants , 2002 .

[15]  Jack Morava.  An algebraic analog of the virasoro group , 2001, math/0109084.

[16]  M. Kapranov.  Semiinfinite symmetric powers , 2001, math/0107089.

[17]  D. Osipov.  Adele constructions of direct images of differentials and symbols , 1997 .

[18]  D V Osipov.  Adele constructions of direct images of differentials and symbols , 1997 .

[19]  J. Brylinski,et al.  The geometry of two-dimensional symbols , 1996 .

[20]  C. Contou-Carrère.  JACOBIENNE LOCALE, GROUPE DE BIVECTEURS DE WITT UNIVERSEL, ET SYMBOLE MODERE , 1994 .

[21]  L. Breen.  On the classification of 2-gerbes and 2-stacks , 1994 .

[22]  P. Deligne.  Le symbole modéré , 1991 .

[23]  Kazuya Kato.  Generalized Class Field Theory , 1990 .

[24]  K. Iwasawa.  Local Class Field Theory , 1986 .

[25]  Kazuya Kato,et al.  Two Dimensional Class Field Theory , 1983 .

[26]  加藤 和也.  A generalization of local class field theory by using K-groups I, II, III = K-群による局所類体論の一般化 , 1980 .

[27]  Jean-Pierre Serre,et al.  Local Class Field Theory , 1979 .

[28]  Kazuya Kato.  A generalization of local class field theory by using $K$-groups, II , 1977 .