It seems to me that there should exist a generalization of the Weil reciprocity law on curves, where instead of functions one takes arbitrary sections of two line bundles. More precisely, it might look like

"For any two line bundles $L$ and $K$ on a curve $C$ and for any two their sections $l$ and $k$ one has

$$\frac{k(\operatorname{div}(L))}{l(\operatorname{div}(K))}=\exp{i\pi E(L,K)}."$$

Here $E(L,K)$ should be some bilinear antisymmetric form on the Jacobian, possibly, a Kahler form. For a function $f$ and a bundle $L$ such that $\operatorname{div}{L}=\sum n_i P_i$ we put $$f(\operatorname{div}(L))=\prod f(P_i)^{n_i}.$$ If $L$ and $K$ are trivial, we get the Weil reciprocity law. The question is how to define $k(\operatorname{div}(L))$ in general so that some generalization of Weil reciprocity will hold true.