The Weil pairing of an isogeny on Galois coefficient modules #
Let φ : W₁ → W₂ be an isogeny of elliptic curves over a field K, of degree n invertible in
K, with dual φ̂ : W₂ → W₁. Its kernels over a separable closure Kˢ are the discrete
G_K-modules E₁(Kˢ)[φ] and E₂(Kˢ)[φ̂] (TauCeti.Isogeny.Hom.KerCoeff), and the Weil pairing
e_n of W₁ induces the isogeny Weil pairing
e_φ : E₁(Kˢ)[φ] × E₂(Kˢ)[φ̂] → μ_n,
characterized by e_φ(P, φ R) = e_n(P, R) for R ∈ E₁(Kˢ)[n]. Every point of E₂(Kˢ)[φ̂] is of
the form φ R (TauCeti.Isogeny.torsionCoeffToDualKerCoeff_surjective), and the value does not
depend on the choice of R because E₁(Kˢ)[φ] is isotropic for e_n. The pairing is
G_K-equivariant and nondegenerate on both sides, so it identifies E₂(Kˢ)[φ̂] with the internal
dual Hom(E₁(Kˢ)[φ], μ_n) as discrete Galois modules (convention Q ↦ (P ↦ e_φ(P, Q))). In
particular the two kernels have the same order.
The resulting cup product pairs H¹(K, E₁[φ]) with H¹(K, E₂[φ̂]) into H²(K, μ_n); on cocycles
it is (g, h) ↦ e_φ(a(g), g • b(h)). This is the pairing under which the local conditions of the
Selmer groups Sel_φ and Sel_φ̂ are compared in Cassels' proof of the isogeny invariance of the
Birch–Swinnerton-Dyer quotient; local Tate duality for it is
TauCeti.Isogeny.localTateKerWeilPairing_bijective.
Main definitions #
TauCeti.Isogeny.kerCoeffWeilPairing: the pairinge_φ : E₁(Kˢ)[φ] × E₂(Kˢ)[φ̂] → μ_n.TauCeti.Isogeny.kerCoeffWeilDual: the equivariant mapE₂(Kˢ)[φ̂] → Hom(E₁(Kˢ)[φ], μ_n).TauCeti.Isogeny.kerH1WeilPairing: the cup productH¹(K, E₁[φ]) × H¹(K, E₂[φ̂]) → H²(K, μ_n)alonge_φ.
Main results #
TauCeti.Isogeny.torsionCoeffWeilPairing_eq_zero_of_pointCoeffMap_eq_zero:E₁(Kˢ)[φ]is isotropic fore_n.TauCeti.Isogeny.kerCoeffWeilPairing_torsionCoeffToDualKerCoeff:e_φ(P, φ R) = e_n(P, R), andTauCeti.Isogeny.kerCoeffWeilPairing_unique: this determinese_φ.TauCeti.Isogeny.kerCoeffWeilPairing_smul:e_φis Galois-equivariant.TauCeti.Isogeny.kerCoeffWeilPairing_nondegenerate_leftandTauCeti.Isogeny.kerCoeffWeilPairing_nondegenerate_right:e_φis nondegenerate.TauCeti.Isogeny.natCard_kerCoeff_dual:E₂(Kˢ)[φ̂]andE₁(Kˢ)[φ]have the same order.TauCeti.Isogeny.kerCoeffWeilDual_bijective:E₂(Kˢ)[φ̂] ≅ Hom(E₁(Kˢ)[φ], μ_n).TauCeti.Isogeny.kerH1WeilPairing_eq_explicitCup11: the cup product on cocycles.
Implementation notes #
Over a separably closed field the corresponding pairing on points is
TauCeti.Isogeny.kerWeilPairing. Only its isotropy input is transported to Kˢ; the coefficient
pairing is constructed directly from WeierstrassCurve.torsionCoeffWeilPairing, which supplies its
Galois equivariance. Nondegeneracy in the second argument is obtained by counting: nondegeneracy in
the first argument, for φ and for φ̂, bounds the order of each kernel by that of the other.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, III.8 and X.4.
- 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.
- J. S. Milne, Arithmetic Duality Theorems, 2nd ed., I, §2.
Isotropy of the kernel #
The kernel of an isogeny is isotropic for the Weil pairing at its degree: if
φ P = φ R = 0 for P, R ∈ E₁(Kˢ)[n], n = deg φ, then e_n(P, R) = 0.
The isogeny Weil pairing #
The isogeny Weil pairing on the Galois coefficient modules, the pairing
e_φ : E₁(Kˢ)[φ] × E₂(Kˢ)[φ̂] → μ_n for n = deg φ invertible in K. It is characterized by
e_φ(P, φ R) = e_n(P, R) for R ∈ E₁(Kˢ)[n]
(TauCeti.Isogeny.kerCoeffWeilPairing_torsionCoeffToDualKerCoeff).
Instances For
The characteristic formula of the isogeny Weil pairing: e_φ(P, φ R) = e_n(P, R) for
P ∈ E₁(Kˢ)[φ] and R ∈ E₁(Kˢ)[n], n = deg φ.
The characteristic formula determines the isogeny Weil pairing.
The isogeny Weil pairing is Galois-equivariant: e_φ(σ P, σ Q) = σ (e_φ(P, Q)) for every
σ ∈ G_K.
The isogeny Weil pairing is nondegenerate in its first argument: if e_φ(P, Q) = 0 for
every Q ∈ E₂(Kˢ)[φ̂], then P = 0.
The kernels of an isogeny and of its dual have the same order over Kˢ, for a degree
invertible in K.
The isogeny Weil pairing is nondegenerate in its second argument: if e_φ(P, Q) = 0 for
every P ∈ E₁(Kˢ)[φ], then Q = 0.
Duality of the kernels as Galois modules #
The isogeny Weil pairing identifies a point Q ∈ E₂(Kˢ)[φ̂] with the character
P ↦ e_φ(P, Q) of E₁(Kˢ)[φ]. The internal hom carries the conjugation action, so this is an
equivariant map even when Q is not rational.
Instances For
Evaluation of the character attached to Q is the isogeny Weil pairing with Q.
The Weil dual of the projection E₁[n] → E₂[φ̂] is restriction of characters along
E₁[φ] → E₁[n].
Duality of the kernels of an isogeny and of its dual as Galois modules: every character
E₁(Kˢ)[φ] → μ_n is P ↦ e_φ(P, Q) for a unique Q ∈ E₂(Kˢ)[φ̂].
The cup product #
The cup product along the isogeny Weil pairing,
H¹(K, E₁[φ]) × H¹(K, E₂[φ̂]) → H²(K, μ_n). It is obtained from the evaluation duality map by the
identification of E₂(Kˢ)[φ̂] with the dual of E₁(Kˢ)[φ].
Instances For
The isogeny Weil cup is the evaluation duality map after applying the dual coefficient map.
The pairing is the usual cup product along e_φ, in the order e_φ(a(g), g • b(h)).
The isogeny Weil cup product with a class projected from degree-torsion is the multiplication Weil cup product with the first class included into degree-torsion.