The kernels of an isogeny, of its dual, and of their degree #
Let φ : W₁ → W₂ be an isogeny of elliptic curves over a field K, with dual φ̂ : W₂ → W₁ and
degree m = deg φ, so that φ̂ ∘ φ = [m]. A point of E₁(Kˢ) killed by φ is killed by m, and
φ sends E₁(Kˢ)[m] into the kernel of φ̂. When φ is separable it is onto on points over
Kˢ, and this gives the short exact sequence of discrete G_K-modules
0 ⟶ E₁(Kˢ)[φ] ⟶ E₁(Kˢ)[m] ⟶ E₂(Kˢ)[φ̂] ⟶ 0.
This file records its two consequences in degree one, which relate the cohomology of the kernels
of φ, of [m] and of φ̂, and so the Selmer groups Sel_φ, Sel_m and Sel_φ̂:
- the sequence
H¹(G_K, E₁[φ]) → H¹(G_K, E₁[m]) → H¹(G_K, E₂[φ̂])is exact; - the kernel of
H¹(G_K, E₁[φ]) → H¹(G_K, E₁[m])consists of the Kummer classes of the points ofE₂(K)killed byφ̂. Indeed the connecting mapH⁰(G_K, E₂[φ̂]) → H¹(G_K, E₁[φ])is the Kummer map ofφonE₂(K)[φ̂], since a preimage underφof a point ofE₂(K)[φ̂]is automaticallym-torsion.
Main definitions #
TauCeti.Isogeny.kerCoeffToTorsionCoeff: the inclusionE₁(Kˢ)[φ] → E₁(Kˢ)[deg φ].TauCeti.Isogeny.torsionCoeffToDualKerCoeff: the mapE₁(Kˢ)[deg φ] → E₂(Kˢ)[φ̂]induced byφ.TauCeti.Isogeny.torsionShortExact: for separableφ, the short exact sequence0 → E₁(Kˢ)[φ] → E₁(Kˢ)[deg φ] → E₂(Kˢ)[φ̂] → 0.TauCeti.Isogeny.finite_kerCoeff_ofIsogeny:E₁(Kˢ)[φ]is finite.
Main results #
TauCeti.Isogeny.degree_nsmul_kerCoeff_eq_zero:E₁(Kˢ)[φ]is killed bydeg φ.TauCeti.Isogeny.range_explicitCoeff1_kerCoeffToTorsionCoeff: exactness ofH¹(G_K, E₁[φ]) → H¹(G_K, E₁[deg φ]) → H¹(G_K, E₂[φ̂]).TauCeti.Isogeny.explicitCoeff1_kerCoeffToTorsionCoeff_eq_zero_iff: a class ofH¹(G_K, E₁[φ])dies inH¹(G_K, E₁[deg φ])exactly when it is the Kummer class of a point ofE₂(K)killed byφ̂.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, III.6.1 and X.4.
φ̂ ∘ φ = [deg φ] on the points over Kˢ.
The coefficient maps #
The inclusion E₁(Kˢ)[φ] → E₁(Kˢ)[deg φ]: a point killed by φ is killed by
deg φ, since φ̂ ∘ φ = [deg φ].
Instances For
The inclusion E₁(Kˢ)[φ] → E₁(Kˢ)[deg φ] does not move points of E₁(Kˢ).
The kernel of an isogeny is killed by its degree: deg φ • P = 0 for P ∈ E₁(Kˢ)[φ],
since φ̂ ∘ φ = [deg φ].
E₁(Kˢ)[φ] is finite: it lies in the finite group E₁(Kˢ)[deg φ]. Unlike
TauCeti.Isogeny.Hom.finite_kerCoeff, which covers every nonzero morphism, this needs no input
beyond the degree. It is an instance because the continuity of the Galois action on the internal
dual Hom(E₁(Kˢ)[φ], μ_n) is inferred from it.
The map E₁(Kˢ)[deg φ] → E₂(Kˢ)[φ̂] induced by φ: the image under φ of a point killed
by deg φ is killed by φ̂, since φ̂ ∘ φ = [deg φ].
Instances For
The map E₁(Kˢ)[deg φ] → E₂(Kˢ)[φ̂] is φ on points of E₁(Kˢ).
H¹(G_K, E₁[φ]) → H¹(G_K, E₁[deg φ]) → H¹(G_K, E₁) is the map induced by the inclusion
E₁(Kˢ)[φ] → E₁(Kˢ).
H¹(G_K, E₁[deg φ]) → H¹(G_K, E₂[φ̂]) → H¹(G_K, E₂) is the map induced by φ on
H¹(G_K, E₁), after the map induced by the inclusion E₁(Kˢ)[deg φ] → E₁(Kˢ).
The short exact sequence of a separable isogeny #
E₁(Kˢ)[deg φ] → E₂(Kˢ)[φ̂] is onto for a separable isogeny φ: a preimage under φ of a
point killed by φ̂ is killed by φ̂ ∘ φ = [deg φ].
The short exact sequence 0 → E₁(Kˢ)[φ] → E₁(Kˢ)[deg φ] → E₂(Kˢ)[φ̂] → 0 of a separable
isogeny φ (Silverman X.4).
Instances For
The inclusion of torsionShortExact φ is TauCeti.Isogeny.kerCoeffToTorsionCoeff.
The projection of torsionShortExact φ is TauCeti.Isogeny.torsionCoeffToDualKerCoeff.
H¹(G_K, E₁[φ]) → H¹(G_K, E₁[deg φ]) → H¹(G_K, E₂[φ̂]) is exact for a separable isogeny
φ.
The connecting map of torsionShortExact φ is the Kummer map of φ: on an invariant of
E₂(Kˢ)[φ̂] coming from Q ∈ E₂(K), the connecting map H⁰(G_K, E₂[φ̂]) → H¹(G_K, E₁[φ]) takes
the value φ.kummerMap Q.
The kernel of H¹(G_K, E₁[φ]) → H¹(G_K, E₁[deg φ]) consists of the Kummer classes of the
points of E₂(K) killed by φ̂, for a separable isogeny φ.