The fixed field of a finite group of translations #
The point group of an elliptic curve W over F acts faithfully on the function field F(W) by
the pullbacks τ_P^* of the translations τ_P : Q ↦ Q + P. This file develops the Galois theory
of that action: a subgroup Φ of points gives a group translationSubgroup of F-algebra
automorphisms of F(W), isomorphic to Φ, and the functions it fixes form the intermediate field
translationFixedField.
For a finite Φ the extension F(W) / F(W) ^ Φ is Galois of degree #Φ, and the
correspondence is exact: every automorphism of F(W) fixing F(W) ^ Φ is a translation by a
point of Φ, so Φ is recovered from its fixed field and distinct finite subgroups have distinct
fixed fields. Read in the other direction, an intermediate field of finite degree is fixed by only
finitely many translations, at most its degree; a subgroup of points whose fixed field has finite
degree is therefore finite, of exactly that order.
This is the field-theoretic half of the construction of the dual isogeny. That construction reads
a separable isogeny φ : W₁ → W₂ of degree n after base change to a separable closure Fˢᵉᵖ,
over which its kernel is a subgroup Φ of n points of W₁(Fˢᵉᵖ): there Fˢᵉᵖ(W₁) is Galois
over the pulled-back copy φ^*Fˢᵉᵖ(W₂) with Φ acting by translations, and identifying
φ^*Fˢᵉᵖ(W₂) as the fixed field of Φ is what factors [n] through φ. Over a base field that
is not separably closed the kernel of φ need not be F-rational, so the subgroup of points of
W(F) used below is in general smaller than that geometric kernel. The degree count and the
correspondence below are what such an identification is read off from; they are stated for an
arbitrary finite subgroup of points, no isogeny being needed to state or prove them.
Main definitions #
WeierstrassCurve.Affine.translationSubgroup: the group of automorphismsτ_P^*ofF(W)forPin a subgroupΦof the point group.WeierstrassCurve.Affine.translationFixedField: the intermediate field of functions fixed by translation by every point ofΦ.WeierstrassCurve.Affine.translationFixingSubgroup: the points whose translation fixes a given intermediate field pointwise.
Main results #
WeierstrassCurve.Affine.translationMulEquiv: the translations byΦare a copy ofΦ.WeierstrassCurve.Affine.finrank_translationFixedFieldandWeierstrassCurve.Affine.isGalois_translationFixedField: for a finiteΦthe function field is Galois of degree#Φover the fixed field.WeierstrassCurve.Affine.fixingSubgroup_translationFixedField: the automorphisms fixing that field are exactly the translations by points ofΦ, whenceWeierstrassCurve.Affine.translationFixingSubgroup_translationFixedFieldandWeierstrassCurve.Affine.eq_of_translationFixedField_eq.WeierstrassCurve.Affine.card_translationFixingSubgroup_leandWeierstrassCurve.Affine.finite_of_finiteDimensional_translationFixedField: a degree bounds the translations that fix it.WeierstrassCurve.Affine.card_translationFixingSubgroup_dvd_finSepDegree: the order of the subgroup fixing an intermediate field of finite degree divides the separable degree above it.WeierstrassCurve.Affine.card_translationFixingSubgroup_eq_finrank_iff: that order is the full degree exactly when the subgroup cuts the field back out.card_translationFixingSubgroup_eq_finrank_iff_isGalois_and_forall_exists_translation: equivalently, the extension is Galois and every automorphism over the field is a translation, so the two conditions together are necessary as well as sufficient.
References #
The group of translations by a subgroup of points, as a subgroup of the F-algebra
automorphisms of the function field: the image of Φ under the translation action.
Equations
Instances For
An automorphism lies in translationSubgroup exactly when it is a translation by a point of
Φ.
Translation by a point of Φ lies in the group of translations by Φ.
A larger subgroup of points translates by a larger group of automorphisms.
Translation identifies a subgroup of points with its group of translations. The action is
faithful, so Φ maps isomorphically onto its image; the point group is written multiplicatively
because the automorphism group is.
Equations
Instances For
The isomorphism of a subgroup of points with its translations is translation.
A subgroup of points has as many translations as it has points.
The fixed field of a group of translations: the functions unmoved by translation by every
point of Φ.
Equations
Instances For
A function lies in the fixed field exactly when no translation by a point of Φ moves
it.
A larger subgroup of points fixes a smaller field.
The function field is Galois over the fixed field of a finite group of translations.
The function field has degree #Φ over the fixed field of a finite subgroup Φ of
points.
Every automorphism of the function field fixing the fixed field of a finite subgroup of
points is a translation by a point of that subgroup. This is the surjectivity half of the Galois
correspondence for the translation action: it is what recognises the extension cut out by Φ as
having no automorphisms beyond the translations.
The points whose translation fixes an intermediate field pointwise.
Equations
Instances For
A point fixes an intermediate field exactly when its translation moves no function of that field.
A larger field is fixed by fewer points.
The two constructions are adjoint: a subgroup of points fixes an intermediate field exactly when that field consists of functions fixed by the subgroup.
A field is fixed by the translations that fix it.
A finite subgroup of points is recovered from its fixed field. Together with
WeierstrassCurve.Affine.le_translationFixedField_translationFixingSubgroup this is the Galois
correspondence for the translation action, in the direction the finite subgroups control.
An intermediate field of finite degree is fixed by at most that many translations. For the
fixed field of a finite subgroup of points the bound is attained, by
WeierstrassCurve.Affine.finrank_translationFixedField.
A subgroup of points whose fixed field has finite degree is itself finite. Its order is
then that degree, by WeierstrassCurve.Affine.finrank_translationFixedField: a subgroup of
translations is no larger than the extension it cuts out.
Distinct finite subgroups of points have distinct fixed fields. Only Φ need be assumed
finite: a subgroup with the same fixed field as a finite one is finite by
WeierstrassCurve.Affine.finite_of_finiteDimensional_translationFixedField.
The order of the fixing subgroup of L divides the separable degree of K(W) over L.
K(W) is Galois, hence separable, over the field the subgroup cuts out, and that relative degree is
the order of the subgroup; it is therefore one factor of the separable degree over L.
The fixing subgroup of L has order the degree of K(W) over L exactly when it cuts L
back out. One inclusion holds for every L, so the cardinality statement and the reverse
inclusion are two names for the same thing: a single field-theoretic statement to aim at in place
of a cardinality one.
If K(W) is finite Galois over L and every automorphism fixing L is a translation, then
L is cut out by exactly as many translations as its degree. With
card_translationFixingSubgroup_eq_finrank_iff this is the reverse inclusion, so it reduces a
field-theoretic question to a group-theoretic one: given that the extension is Galois, are there
automorphisms over L beyond the translations?
The fixing subgroup of L has order the degree exactly when K(W) is Galois over L and
every automorphism over L is a translation. The two hypotheses of
card_translationFixingSubgroup_eq_finrank_of_forall_exists_translation are therefore jointly
necessary as well as sufficient.