Weil pairing
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In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing (bilinear form, though with multiplicative notation) on the torsion subgroup of E. The name is for André Weil, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.
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[edit] Formulation
Suppose E is defined over a field K. Given an integer n > 0 (we require n to be prime to char(K) if char(K)> 0) such that K contains a primitive nth root of unity, then the n-torsion on 
has known structure, as a Cartesian product of two cyclic groups of order n. The basis of the construction is of an n-th root of unity
for given points ![P,Q \in E(K)[n]](http://upload.wikimedia.org/math/7/9/1/791a6fc99204b2640076061b91434669.png)
, where ![E(K)[n]=\{T \in E(K) \mid n \cdot T = O \}](http://upload.wikimedia.org/math/e/6/8/e687ffd4ba143c09c65ef7ae8c4cfee7.png)
and 
, by means of Kummer theory.
By a direct argument one can define a function F in the function field of E over the algebraic closure of K, by its divisor:
with sums for 0 ≤ k < n. In words F has a simple zero at each point P + kQ, and a simple pole at each point kQ. 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. One can show that
unless P and Q generate cyclic subgroups one of which is inside the other. In fact then G/F would yield a function on the isogenous curve E/C where C is the cyclic subgroup generated by Q, having just one simple pole. Such a function cannot exist, as follows by proving the residue at the pole is zero, a contradiction.
Therefore if we define
we shall have an n-th root of unity (translating n times must give 1) other than 1. With this definition it can be shown that w is antisymmetric and bilinear, giving rise to a non-degenerate pairing on the n-torsion.
[edit] Generalization to abelian varieties
For abelian varieties over an algebraically closed field k, the Weil pairing is a nondegenerate pairing
![A[n] \times A^\vee[n] \longrightarrow \mu_n](http://upload.wikimedia.org/math/4/8/9/4890bd0b6214ccee1570bcbf6663100e.png)
for all n prime to the characteristic of k.[1]. Here 
denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarization
,
then composition gives a (possibly degenerate) pairing
![A[n] \times A[n] \longrightarrow \mu_n.](http://upload.wikimedia.org/math/8/9/c/89ca42d092f6ed3db4274862397caa77.png)
If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarization of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians). Hence, composing the Weil pairing for J with the polarization gives a nondegenerate pairing
![J[n]\times J[n] \longrightarrow \mu_n](http://upload.wikimedia.org/math/5/3/1/531b864742658fc704f609e2968b2bfe.png)
for all n prime to the characteristic of k.
As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C.
[edit] Applications
The Weil pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.
[edit] See also
[edit] References
- ^ James Milne, Abelian Varieties, available at www.jmilne.org
