Tropical Weil's reciprocity law and Weil's pairing

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.