The statement is obviously false in general (e.g. if P=Q).
- That should be covered by expanding the remark about Weierstrass sigma-functions. Otherwise it's really just the remark that once you choose a free basis in such a way that you get a pairing of 1, the (elliptic curve case) pairing is all defined. (I've fixed up the degenerate cases.) Charles Matthews (talk) 19:31, 15 February 2009 (UTC)
In current version (25 01 2012) line 3 in section "Formulation" seems to be unclear and incorrect.
As far as I understand n-torsion points on Jacobian are more or less etale-(co?)homology of the curve. So Weil pairing can be seen product in cohomology (or dually intersection in homology)... If any one can make this precise would be worth ! — Preceding unsigned comment added by Alexander Chervov (talk • contribs) 18:33, 26 January 2012 (UTC)
"then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism"
The above suggests that principal polarizations may not be isomorphisms. Principal polarizations are always isomorphisms. The theta-divisor is "special" because it always gives a principal polarization. — Preceding unsigned comment added by 188.8.131.52 (talk) 21:00, 25 January 2012 (UTC)