Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids Over an Artinian Local Ring

Abstract

Abstract. The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-DeConcini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

On the Kernel and the Image of the Rigid Analytic Regulator in Positive Characteristic

“…was defined, called the Contou-Carrère symbol in [1], where t is a variable. The Contou-Carrère symbol is equal to the tame symbol if A is a field.…”

Generalized reciprocity laws

“…Here we have used the theory of the Sato Grassmannian and determinant bundles to construct the central extensions. Of course, one can construct these extensions using a more elementary language, as is done for example in [3]. The reason for using these more sophisticated techniques is that in this way the results can be generalized to the case of arithmetic curves and, in general, curves defined over a ring R. We hope to develop this theory over arbitrary Noetherian rings elsewhere.…”

A new explicit expression of the Contou-Carrère symbol

“…The symbol ·, · A is clearly antisymmetric and, although it is not immediately obvious from the definition, also bimultiplicative. G. W. Anderson and the author [1] have interpreted the Contou-Carrère symbol f, g A -up to signs-as a commutator of liftings of f and g to a certain central extension of a group containing A((t)) × , and they have exploited the commutator interpretation to prove, in the style of Tate [11], a reciprocity law for the ContouCarrère symbol on a nonsingular complete curve defined over an algebraically closed field k, A being an artinian local k-algebra.…”

“…readers are referred to [1]. For arbitrary elements f, g, h ∈ A((t)) × , the following relations hold:…”

“…Accordingly, if Tr A denotes the trace of a square matrix A, bearing in mind that When X is a complete curve, a direct consequence of the reciprocity law for the Contou-Carrère symbol [1] is the following Residue Theorem: …”