2004
DOI: 10.1081/agb-120027853
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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 class… Show more

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Cited by 35 publications
(43 citation statements)
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“…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.…”
Section: Introduction and Announcement Of Resultsmentioning
confidence: 99%
“…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.…”
Section: Introduction and Announcement Of Resultsmentioning
confidence: 99%
“…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.…”
Section: José M Muñoz Porras and Fernando Pablos Romomentioning
confidence: 99%
“…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.…”
Section: Let F G ∈ A((t))mentioning
confidence: 99%
“…readers are referred to [1]. For arbitrary elements f, g, h ∈ A((t)) × , the following relations hold:…”
Section: A[[t]] A((t))mentioning
confidence: 99%
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