A Note on Steinberg Symbols on Algebraic Curves

Abstract

The aim of this paper is to provide a method for defining Steinberg symbols on a complete algebraic curve over a perfect field k from the commutator of a certain extension of groups. This extension is associated with a group morphism ϕ : k* → G. With this definition the reciprocity law is a consequence of the finiteness of the cohomology groups H 0(C, 𝒪 C ) and H 1(C, 𝒪 C ). Using this method, Hilbert's norm residue symbol on an algebraic curve and the symbol (a, b) v for the field ℚ p (n = 2) can be defined.