Integer endomorphisms for a transcendental j-invariant

Over an algebraically closed field of characteristic zero, a finite integral self-embedding of an elliptic function field induces a point-group endomorphism via divisor pushforward and Pic0. WeierstrassCurve.isogenyPointEnd constructs this endomorphism, including the point/place correspondence, Abel's theorem and the norm formula. None of these auxiliary structures is an assumption of the public interface.

WeierstrassCurve.isogenyEnd_eq_zsmul_of_transcendental_j proves that this endomorphism is integer multiplication when the j-invariant is transcendental over ℚ. This is a statement about the endomorphisms represented by Affine.IsogenyEndDatum, not about arbitrary abstract group endomorphisms of the points, nor a comparison with a separate scheme-theoretic endomorphism ring.

noncomputable def WeierstrassCurve.isogenyPointEnd {F : Type u_1} [Field F] [DecidableEq F] [IsAlgClosed F] [CharZero F] (W : WeierstrassCurve F) [W.IsElliptic] (D : W.toAffine.IsogenyEndDatum) : AddMonoid.End W.toAffine.Point

The point-group endomorphism induced by a finite integral self-embedding of the function field, with the divisor-theoretic infrastructure constructed rather than assumed.

Equations

• W.isogenyPointEnd D = D.pointEnd ⋯

theorem WeierstrassCurve.isogenyEnd_eq_zsmul_of_transcendental_j {F : Type u_1} [Field F] [DecidableEq F] [IsAlgClosed F] [CharZero F] (W : WeierstrassCurve F) [W.IsElliptic] (ht : Transcendental ℚ W.j) (D : W.toAffine.IsogenyEndDatum) : ∃ (m : ℤ), ∀ (P : W.toAffine.Point), (W.isogenyPointEnd D) P = m • P

An elliptic curve with transcendental j-invariant has only integer endomorphisms among the point-group endomorphisms induced by finite integral self-embeddings of its function field.