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I was once told, by someone who would likely be right about such things, that the version of Weil reciprocity for abelian varieties (as in Lang, Abelian Varieties) should come out of consideration of biextensions. I'd be quite intrigued to know if this has ever been worked out and written down.

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Lang's reciprocity law and Poincar\'e-Mumford biextensions come together in the context of generalized N\'eron pairings

http://iopscience.iop.org/0025-5726/6/3/A03/pdf/0025-5726_6_3_A03.pdf

(Proof of Proposition 2 on p. 496 and p. 502).

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