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

Greg W. Anderson, Fernando Pablos Romo

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

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-De Concini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

Original languageEnglish (US)
Pages (from-to)79-102
Number of pages24
JournalCommunications in Algebra
Volume32
Issue number1
DOIs
StatePublished - Jan 2004

Bibliographical note

Funding Information:
The second-named author was partially supported by DGESYC research contract BFM2000-1327 and Castilla y León regional government contract SA064/01.

Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

Keywords

  • Contou-Carrère
  • Determinant groupoids
  • Explicit reciprocity laws

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