3-Cocycles, symbols and reciprocity laws on curves

We introduce a new approach for the study of two-dimensional symbols, F*×F*×F*→GF*×F*×F*→G, where FF is a discrete valuation field and G   is a commutative group. From central extensions of groups we obtain a three-cocycle {·,·,·}vF,vKϕ and the symbol is a differentiated element of the cohomology class [{·,·,·}vF,vKϕ]∈H3(F*,G). Our construction generalizes well-known two-dimensional symbols, such as the Parshin symbol on a surface, and we offer a proof and a conjecture for reciprocity laws on curves related to these symbols.