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\'eMumford biextensions come together in the context of generalized N\'eron pairings
http://iopscience.iop.org/00255726/6/3/A03/pdf/00255726_6_3_A03.pdf
(Proof of Proposition 2 on p. 496 and p. 502).

1$\begingroup$ NÉRON PAIRING AND QUASICHARACTERS 1972 Math. USSR Izv. 6 491 Access restricted: you will need to login or make a payment to access the full text of this article. $\endgroup$ – Chandan Singh Dalawat Jun 8 '12 at 8:40

2$\begingroup$ Sorry to hear that. However, the original Russian version is available for free (see pp. 502 and 508509 of) mathnet.ru/php/… mathnet.ru/links/07870c75a71989615a88f79957d8d509/im2308.pdf $\endgroup$ – Yuri Zarhin Jun 8 '12 at 10:06