Tropical Weil's reciprocity law and Weil's pairing

TL;DR

This paper develops a tropical analogue of Weil's reciprocity law for meromorphic functions on complex curves, providing a combinatorial proof and constructing a tropical Weil pairing on degree-zero divisors.

Contribution

It introduces a tropical version of Weil's reciprocity law and constructs a tropical Weil pairing, offering new combinatorial insights into classical algebraic geometry.

Findings

Tropical Weil reciprocity law is established.

A combinatorial proof of classical Weil reciprocity is provided.

A tropical Weil pairing on divisors of degree zero is constructed.

Abstract

The Weil reciprocity law asserts that given two meromorphic functions $f, g$ on a compact complex curve, the product of the values of $f$ over the roots and poles of $g$ is equal to the product of the values of $g$ over the roots and poles of $f$. We state and prove a tropical version of this reciprocity; the tropical ideas lead to yet another transparent ``combinatorial'' proof of the classical Weil reciprocity law. Then, we construct a tropical Weil pairing on the set of divisors of degree zero.