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Simple Proofs of Classical Explicit Reciprocity Laws on Curves Using Determinant Groupoids Over an Artinian Local Ring#.
- Source :
-
Communications in Algebra . Jan2004, Vol. 32 Issue 1, p79-102. 24p. - Publication Year :
- 2004
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00927872
- Volume :
- 32
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Communications in Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 12453672
- Full Text :
- https://doi.org/10.1081/AGB-120027853