### Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an artinian local ring, by Greg W. Anderson and Fernando Pablos Romo

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-Carr\`{e}re 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-DeConcini-Kac on Weil reciprocity. We also build in an
essential way on a previous work of the second-named author.

Greg W. Anderson and Fernando Pablos Romo <gwanders@math.umn.edu, fpablos@usal.es>