The Weil pairing on the kernels of dual isogenies #
For an isogeny φ : W₁ → W₂ of degree n invertible in a separably closed field F,
there is a nondegenerate bilinear pairing
ker φ × ker φ̂ → Additive (rootsOfUnity n F).
It is characterized by e_φ(P, φ R) = e_n(P, R) for R ∈ W₁[n]. Every point of
ker φ̂ has such a lift. The value is independent of the lift because ker φ is
isotropic for e_n: lift its first argument through φ̂ and use the adjointness of
φ and φ̂. This kernel pairing is the coefficient pairing used in the local duality
of isogeny descent, unlike the restriction of e_n to ker φ × ker φ, which is zero.
The construction uses the native point maps, the existing Weil pairing and its adjointness
under dual isogenies. No choice of bases or cyclicity of either kernel is required.
The maps kerToTorsion and torsionToDualKer are the native-point counterparts of
TauCeti.Isogeny.kerCoeffToTorsionCoeff and TauCeti.Isogeny.torsionCoeffToDualKerCoeff;
TauCeti.Isogeny.torsionShortExact in
TauCeti.AlgebraicGeometry.EllipticCurve.GaloisCohomology.Isogeny.Torsion packages their
sequence as discrete Galois modules. The pairing is defined on geometric points;
no Galois action is considered.
References #
- J. H. Silverman, The Arithmetic of Elliptic Curves, III.8, the isogeny Weil pairing.
- J. W. S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. reine angew. Math. 217 (1965), 180–199.
The kernel of an isogeny is isotropic for the Weil pairing at its degree.
The isogeny Weil pairing on the kernels of φ and φ̂. It is characterized by
e_φ(P, φ R) = e_n(P, R) on degree torsion, where n = deg φ.
Instances For
The characteristic formula for the kernel pairing, without choosing a lift.
The characteristic formula determines the bilinear kernel pairing uniquely.
The isogeny Weil pairing is nondegenerate in its first argument.
The isogeny Weil pairing is nondegenerate in its second argument.