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A general reciprocity law for symbols on arbitrary vector spaces. (English) Zbl 1441.11287

Extending previous work by himself and his co-authors, motivated by the approach of J. Tate [Ann. Sci. Éc. Norm. Supér. (4) 1, No. 1, 149–159 (1968; Zbl 0159.22702)] and of E. Arbarello et al. [Proc. Symp. Pure Math. 49, 171–190 (1989; Zbl 0699.22028)], the author develops a general formalism of symbols, residues and reciprocity laws. He states an elegant general reciprocity law and then shows how it can be used to derive various classical and modern reciprocity laws on curves and surfaces, including Weil’s reciprocity law and more recent results of Osipov, Parshin, Horozov and the author himself.

MSC:

11R56 Adèle rings and groups
15A03 Vector spaces, linear dependence, rank, lineability
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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