Central extensions and reciprocity laws on algebraic surfaces

Abstract

We consider an algebraic surface. For an irreducible curve on this surface and for a point on this curve one can associate an artinian ring, which is a sum of two-dimensional local fields. An example of two-dimensional local field is iterated Laurent series. We consider an integer value symbol on a two-dimensional local field. This symbol appears for the description of non-ramified coverings of two-dimensional local fields and for the description of intersection index of divisors on surfaces by adeles. We interpret this symbol as the commutator of lifting of elements in the central extension of the multiplicative group of two-dimensional local field. We prove the reciprocity laws around the points and along the projective curves on the surface. The proof uses the adelic ring on the surface and is similar to the Tate method for the residue of differentials on a projective curve.

submitted to "Matematicheskiy Sbornik"

References in corpus (2)

• Krichever correspondence for algebraic varieties
• Semiinfinite symmetric powers

Cited by in corpus (8)

• The two-dimensional Contou-Carrère symbol and reciprocity laws
• Representations of Higher Adelic Groups and Arithmetic
• A categorical proof of the Parshin reciprocity laws on algebraic surfaces
• Unramified two-dimensional Langlands correspondence
• Adeles on n-dimensional schemes and categories C_n
• Local Beilinson-Tate Operators
• Arithmetic Central Extensions and Reciprocity Laws for Arithmetic Surface
• Two dimensional versions of the affine Grassmannian and their geometric description