Weil pairing

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In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, 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.

Formulation[edit]

Choose an elliptic curve E defined over a field K, and 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 E(\overline{K}) is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity

w(P,Q) \in \mu_n

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

A down-to-earth construction of the Weil pairing is as follows. Choose a function F in the function field of E over the algebraic closure of K with divisor

 \mathrm{div}(F)= \sum_{0 \leq k < n}(P+k\cdot Q) - \sum_{0 \leq k < n} (k\cdot Q).

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

 w(P,Q):=\frac{G}{F}

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 antisymmetric and bilinear, 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(E) × T(E) → T(μ) on the Tate module T(E) of the elliptic curve E (the inverse limit of the ℓn-torsion points) to the Tate module T(μ) of the multiplicative group (the inverse limit of ℓn roots of unity).

Generalisation to abelian varieties[edit]

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

for all n prime to the characteristic of k.[1] Here A^\vee denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation

\lambda: A \longrightarrow A^\vee,

then composition gives a (possibly degenerate) pairing

A[n] \times A[n] \longrightarrow \mu_n.

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 polarisation 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 polarisation gives a nondegenerate pairing

 J[n]\times J[n] \longrightarrow \mu_n

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.

Applications[edit]

The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.

See also[edit]

References[edit]

  1. ^ James Milne, Abelian Varieties, available at www.jmilne.org/math/

External links[edit]