Jinhyun Park, A Reciprocity Theorem Arising from a Family of Algebraic Curves

The classical reciprocity theorem, also called the residue theorem, states that the sum of the residues of a rational (meromorphic) differential form on a compact Riemann surface is zero. Its generalization to smooth projective curves over a field is often called the Tate reciprocity theorem.

There is a different “multiplicative version” too. Here, instead of a rational form, one uses a pair of rational functions on a smooth projective curve, and instead of residues, one uses “Tame symbols”. The corresponding global result is called the Weil reciprocity. This result is elegantly reformulated in terms of the Milnor K-theory, and it is generalized to sequences of rational functions by A. Suslin. This Suslin reciprocity was recently strengthened by D. Rudenko, resolving a conjecture of A. Goncharov.

In this talk, let me sketch my recent work in-progress, that studies a different kind of reciprocity results coming from a proper family of algebraic curves over an algebraically closed field of characteristic 0.