The units of the endomorphism monoid

An endomorphism of W is invertible exactly when it has degree one. Degree one means the function-field pullback is onto, and the factorisation theorem turns a surjective pullback into an isogeny inverting it on both sides; conversely degree is multiplicative and the identity has degree one, so a unit's degree divides one.

Units is defined for a monoid, so (Hom W W)ˣ is determined by the multiplicative structure alone: the group described here is the same one the endomorphism ring will have.

Main results

• TauCeti.Isogeny.Hom.isUnit_iff_degree_eq_one: a unit is exactly a degree-one endomorphism.

References

• J. Silverman, The Arithmetic of Elliptic Curves, II.2.4.1.

@[simp]

theorem TauCeti.Isogeny.Hom.isUnit_iff_degree_eq_one {F : Type u_1} [Field F] {W₁ : WeierstrassCurve.Affine F} {f : Hom W₁ W₁} : IsUnit f ↔ f.degree = 1

The units of the endomorphism monoid are exactly the degree-one endomorphisms, an isogeny of degree one being an isomorphism. Units depends only on the multiplicative structure, so this describes the whole unit group of the endomorphism monoid.

@[simp]

theorem TauCeti.Isogeny.Hom.degree_coe_units {F : Type u_1} [Field F] {W₁ : WeierstrassCurve.Affine F} (u : (Hom W₁ W₁)ˣ) : (↑u).degree = 1

An automorphism has degree one.