Galois conjugation of isogenies #
Let W₁ and W₂ be Weierstrass curves over F, and let K/F be a field extension. An
F-automorphism σ of K conjugates an isogeny φ : (W₁)_K → (W₂)_K: apply σ⁻¹ to
the coefficients of a function on the target, pull it back along φ, and apply σ to the
resulting function on the source. The two semilinearities cancel, so the conjugate pullback is
again a K-algebra homomorphism.
Conjugation preserves the condition that infinity maps to infinity. Algebraically, a monic integral-dependence relation for the source coordinate is transported by the coefficient automorphisms. The construction therefore gives a genuine Galois action on the type of isogenies, and it respects identity and composition. These are the equivariance facts needed to descend an isogeny constructed after extending the base field.
Main definitions #
TauCeti.CoordinatePullback.galoisConj: conjugation of a coordinate pullback.TauCeti.Isogeny.galoisConj: conjugation of an isogeny.TauCeti.Isogeny.galoisAction: the resulting action on isogenies between two fixed base-changed curves.
Main results #
TauCeti.Isogeny.galoisConj_fieldPullback: conjugation commutes with extension of a pullback to the function fields.TauCeti.Isogeny.galoisConj_eq_iff_fieldPullback_equivariant: conjugation fixes an isogeny exactly when its function-field pullback commutes with the coefficient automorphism.TauCeti.Isogeny.galoisConj_idandTauCeti.Isogeny.galoisConj_comp: conjugation preserves identity and composition.TauCeti.Isogeny.galoisConj_map_algebraMap: an isogeny defined over the ground field is fixed.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, II.2 and III.6.
Galois conjugation of a coordinate pullback. Its underlying ring homomorphism is
σ ∘ φ ∘ σ⁻¹; the semilinearity of the two outer maps cancels, making the composite
K-linear.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The conjugate pullback is obtained by applying σ⁻¹ on the target, then φ, then σ
on the source.
Conjugation by the identity fixes a coordinate pullback.
Successive Galois conjugations multiply their automorphisms.
Galois conjugation preserves the condition that infinity maps to infinity.
Galois conjugation of an isogeny between base changes of curves defined over F.
Equations
- TauCeti.Isogeny.galoisConj W₁ W₂ φ σ = { pullback := TauCeti.CoordinatePullback.galoisConj W₁ W₂ φ.pullback σ, mapsInfinity := ⋯ }
Instances For
The pullback of the conjugate is the conjugate of the pullback.
Conjugating a function-field pullback is the function-field pullback of the conjugate isogeny.
Galois conjugation fixes an isogeny exactly when its function-field pullback commutes with the corresponding coefficient automorphism. This compares actual field maps, including their action on functions with poles.
Conjugation by the identity fixes an isogeny.
Successive Galois conjugations multiply their automorphisms.
Galois conjugation fixes the identity isogeny.
Galois conjugation respects composition of isogenies.
An isogeny defined over the ground field is fixed by Galois conjugation.
The Galois action on isogenies between two fixed base-changed curves.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The bundled Galois action is Galois conjugation.