Galois descent of Weierstrass function-field maps

A function-field map between base changes of two Weierstrass curves descends uniquely to the ground field if and only if it commutes with the coefficient Galois actions. The criterion applies to arbitrary Galois extensions, including a separable closure over an imperfect field; neither ellipticity nor finiteness of the extension is needed.

The descended map is characterized by its commuting square with the two function-field base-change maps. This is the field-map descent step in constructing a dual isogeny over its field of definition. Pointedness and degree are separate properties of that construction.

References

• J. Silverman, The Arithmetic of Elliptic Curves, II.2 and III.6.

theorem WeierstrassCurve.existsUnique_functionFieldMap_iff_galoisEquivariant {F : Type u_1} {K : Type u_2} [Field F] [Field K] [Algebra F K] [IsGalois F K] (W₁ W₂ : WeierstrassCurve F) (g : (W₂.baseChange K).toAffine.FunctionField →ₐ[K] (W₁.baseChange K).toAffine.FunctionField) : (∃! f : W₂.toAffine.FunctionField →ₐ[F] W₁.toAffine.FunctionField, ∀ (z : W₂.toAffine.FunctionField), (Affine.FunctionField.map W₁.toAffine (algebraMap F K)) (f z) = g ((Affine.FunctionField.map W₂.toAffine (algebraMap F K)) z)) ↔ ∀ (σ : Gal(K/F)) (z : (W₂.baseChange K).toAffine.FunctionField), (W₁.functionFieldGaloisAction σ) (g z) = g ((W₂.functionFieldGaloisAction σ) z)

A map between the function fields of two base-changed Weierstrass curves descends uniquely to the ground field exactly when it commutes with every coefficient automorphism. The equality in the existence statement is the defining base-change square of the descended map.