Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring

Details Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring

2002-07-31

arxiv.org/abs/math/0207311v1

Mathematics

Number Theory

Scientific paper

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-Carr\`{e}re 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.

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