Editing Weil pairing (section)

==Formulation==
Choose an elliptic curve ''E'' defined over a [[field (mathematics)|field]] ''K'', and an integer ''n''&nbsp;&gt;&nbsp;0 (we require ''n'' to be coprime to char(''K'') if char(''K'')&nbsp;&gt;&nbsp;0) such that ''K'' contains a [[primitive nth root of unity]]. Then the ''n''-torsion on <math>E(\overline{K})</math> is known to be a [[Cartesian product]] of two [[cyclic group]]s of order ''n''. The Weil pairing produces an ''n''-th root of unity

:<math>w(P,Q) \in \mu_n</math>

by means of [[Kummer theory]], for any two points <math>P,Q \in E(K)[n]</math>, where <math>E(K)[n]=\{T \in E(K) \mid n \cdot T = O \} </math> and <math>\mu_n = \{x\in K \mid x^n =1 \} </math>.

A down-to-earth construction of the Weil pairing is as follows.{{Citation needed|date=December 2024}} Choose a function ''F'' in the [[function field of an algebraic variety|function field]] of ''E'' over the [[algebraic closure]] of ''K'' with [[divisor (algebraic geometry)|divisor]]

:<math> \mathrm{div}(F)= \sum_{0 \leq k < n}[P+k\cdot Q] - \sum_{0 \leq k < n} [k\cdot Q]. </math>

So ''F'' has a simple zero at each point ''P'' + ''kQ'', and a simple pole at each point ''kQ'' if these points are all distinct. Then ''F'' is well-defined up to multiplication by a constant. If ''G'' is the translation of ''F'' by ''Q'', then by construction ''G'' has the same divisor, so the function ''G/F'' is constant.

Therefore if we define

:<math> w(P,Q):=\frac{G}{F}</math>

we shall have an ''n''-th root of unity (as translating ''n'' times must give 1) other than 1. With this definition it can be shown that ''w'' is alternating and bilinear,<ref>{{cite book|last1=Silverman|first1=Joseph|title=The Arithmetic of Elliptic Curves|date=1986|publisher=Springer-Verlag|location=New York|isbn=0-387-96203-4}}</ref> giving rise to a non-degenerate pairing on the ''n''-torsion.

The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of ''n''-torsion points) because the pairings for different ''n'' are not the same. However
they do fit together to give a pairing ''T''<sub>ℓ</sub>(''E'') × ''T''<sub>ℓ</sub>(''E'')  → ''T''<sub>ℓ</sub>(μ) on the [[Tate module]] ''T''<sub>ℓ</sub>(''E'') of the elliptic curve ''E'' (the inverse limit of the ℓ<sup>''n''</sup>-torsion points) to the Tate module ''T''<sub>ℓ</sub>(μ) of the multiplicative group (the inverse limit of ℓ<sup>''n''</sup> roots of unity).