The dual of a separable isogeny over a separably closed field #
Over a separably closed field the kernel of a separable isogeny φ : W₁ → W₂ has exactly deg φ
points (TauCeti.Isogeny.card_ker_eq_degree). So [deg φ] factors through φ by a unique
isogeny (TauCeti.Isogeny.existsUnique_comp_eq_mulByIntIsogenyOfNeZero_degree). This factor is
the dual isogeny φ̂ : W₂ → W₁ (Silverman III.6.1), and this file names it and proves its
basic properties (Silverman III.6.2(a), (c), (d), (e), (f)):
φ̂ ∘ φ = [deg φ]onW₁, andφ̂is the only isogeny with this property;φ ∘ φ̂ = [deg φ]onW₂;deg φ̂ = deg φ;(ψ ∘ φ)^ = φ̂ ∘ ψ̂for separableφ,ψ;φ̂̂ = φwheneverφ̂is itself separable;[n]̂ = [n]whenevernis nonzero in the base field.
The identity φ ∘ φ̂ = [deg φ] needs φ ∘ [n] = [n] ∘ φ
(TauCeti.Isogeny.comp_mulByIntIsogenyOfNeZero). Precomposing with φ is injective, so it cancels
from φ ∘ φ̂ ∘ φ = φ ∘ [deg φ] = [deg φ] ∘ φ.
Main definitions #
TauCeti.Isogeny.dual: the dualφ̂of a separable isogenyφover a separably closed field.
Main results #
TauCeti.Isogeny.dual_compandTauCeti.Isogeny.eq_dual_iff_comp_eq:φ̂ ∘ φ = [deg φ], and this characterisesφ̂.TauCeti.Isogeny.comp_dual:φ ∘ φ̂ = [deg φ].TauCeti.Isogeny.degree_dual:deg φ̂ = deg φ.TauCeti.Isogeny.ofIsogeny_dual_comp_ofIsogenyandTauCeti.Isogeny.ofIsogeny_comp_ofIsogeny_dual: the two composites aredeg φ • 1in the additive groups of morphisms.TauCeti.Isogeny.pointMap_dual_pointMapandTauCeti.Isogeny.pointMap_pointMap_dual: on points,φ̂ (φ P) = deg φ • Pandφ (φ̂ Q) = deg φ • Q.TauCeti.Isogeny.dual_comp_dual:(ψ ∘ φ)^ = φ̂ ∘ ψ̂.TauCeti.Isogeny.dual_dual:φ̂̂ = φwhenφ̂is separable.TauCeti.Isogeny.dual_mulByIntIsogeny:[n]is self-dual when it is separable.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, III.4.8, III.4.10, III.6.1 and III.6.2.
The dual isogeny φ̂ : W₂ → W₁ of a separable isogeny φ : W₁ → W₂ over a separably
closed field: the unique isogeny with φ̂ ∘ φ = [deg φ] (Silverman III.6.1).
Instances For
The dual of φ composed with φ is multiplication by deg φ on W₁
(Silverman III.6.1, III.6.2(a)).
φ̂ is the only isogeny χ with χ ∘ φ = [deg φ].
The dual has the same degree (Silverman III.6.2(e)).
φ composed with its dual is multiplication by deg φ on W₂ (Silverman III.6.2(a)).
φ̂ ∘ φ = deg φ • 1 in the additive group of morphisms of W₁.
φ ∘ φ̂ = deg φ • 1 in the additive group of morphisms of W₂.
On points, φ̂ (φ P) = deg φ • P.
On points, φ (φ̂ Q) = deg φ • Q.
The dual of the dual is the original isogeny, when the dual is separable (Silverman III.6.2(f)).
The dual of a composite is the composite of the duals in the opposite order:
(ψ ∘ φ)^ = φ̂ ∘ ψ̂ (Silverman III.6.2(c)).
Multiplication by n is self-dual when it is separable, that is, when n is nonzero in
F (isSeparable_mulByIntIsogeny_iff; Silverman III.6.2(d)).