Elliptic Curves
Isogenies and Endomorphism Rings
Isogenies as maps of curves respecting the group law, and the two possible endomorphism rings over a field of characteristic zero.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0638
An isogeny is a non-constant map of curves preserving the identity, and it automatically preserves the group law. The endomorphisms of a curve form a ring, and which ring it is has deep arithmetic consequences.
Isogenies
- Definition
- A non-constant morphism of curves taking identity to identity. It is automatically a group homomorphism.
- Degree
- The size of the kernel for a separable isogeny. The kernel is finite.
- Dual isogeny
- Every isogeny has a dual in the opposite direction, and the composition is multiplication by the degree.
- Lattice description
- Over the complex numbers, an isogeny corresponds to an inclusion of one lattice in another with finite index.
Endomorphisms
An endomorphism is an isogeny from a curve to itself, together with the zero map. These form a ring under addition and composition.
| Endomorphism ring | Condition | Frequency |
|---|---|---|
| The integers | Only multiplication-by-n maps | The generic case |
| An order in an imaginary quadratic field | Complex multiplication | Special; countably many j-invariants |
| An order in a quaternion algebra | Supersingular curves; positive characteristic only | Special |
Complex multiplication
A curve has complex multiplication when its endomorphism ring is strictly larger than the integers. Over the complex numbers this means the lattice is preserved by multiplication by a non-real complex number, which forces that number to be an imaginary quadratic algebraic integer.
Computing isogenies
Given a finite subgroup, Velu's formulas construct the quotient curve and the isogeny explicitly. This is the standard computational tool and it underlies isogeny-based constructions throughout.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.2.2-7.2.3. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.