Isogenies and torsion subgroups
Relate finite morphisms, function-field degree and geometric kernels without confusing separable and inseparable behavior.
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The question for today
What you’ll understand
- Define isogeny degree through the induced extension of function fields.
- Distinguish geometric kernel size from total degree in positive characteristic.
- Separate geometric n-torsion from the subgroup rational over the base field.
An isogeny relates elliptic curves through a finite surjective group morphism. Its kernel is finite, but counting its visible points is not always the same as measuring its degree. Positive characteristic introduces inseparability, where multiplicities carry information invisible in the underlying set of points. Torsion presents a second distinction: even when all geometric torsion is understood, only some of those points may be rational over the field used for arithmetic. Both distinctions are essential before using a kernel to classify a map.
A morphism has a function-field degree
An isogeny from E to E-prime over k is a nonconstant morphism sending O to O-prime; it is automatically a surjective group homomorphism. Its degree is the degree of the induced finite extension of function fields. This geometric degree counts fibers with multiplicity and is multiplicative under composition. An isomorphism has degree one. The zero homomorphism is customarily included in the Hom group, but it is not a nonconstant isogeny under the definition used here.
In characteristic zero every finite function-field extension in this setting is separable, so an isogeny's degree equals the number of geometric kernel points. More generally, total degree factors as separable degree times inseparable degree. Fibers of a group morphism are translates of its kernel, and the geometric kernel contains precisely the separable-degree number of points. Over the algebraic closure, inseparable multiplicity can make the scheme-theoretic kernel longer than its set of points.
Multiplication and Frobenius expose the distinction
For a positive integer n, multiplication by n has degree n². When n is prime to the characteristic, the map is separable and its geometric kernel is two copies of the cyclic group of order n. Rational n-torsion is the Galois-fixed subgroup of those geometric points, and can be much smaller. Knowledge of E[n] over an algebraic closure therefore does not establish that E(k) contains n² points of order dividing n.
In characteristic p, relative Frobenius maps E to its p-th coefficient twist by raising coordinates to the p-th power. It has degree p and is purely inseparable, with just O in its geometric kernel. Multiplication by p has degree p², but its geometric kernel has p points in the ordinary case and one point in the supersingular case. These are statements about geometric p-torsion, not claims that every such point is rational over the original finite field.
Explicit quotients and dual maps
Over a perfect field k, a finite subgroup of geometric points stable under the action of the base-field Galois group determines a separable quotient isogeny defined over that field. Stability of the subgroup does not require every individual point to be rational. Explicit formulas such as Vélu's construct the quotient, but their denominators must be interpreted as rational expressions extending to the projective curve. Kernel points are mapped to infinity, rather than discarded because an affine formula has a pole.
Every isogeny has a dual isogeny in the reverse direction whose compositions in either order are multiplication by the original degree. Thus an isogeny has an inverse up to a multiplication map, not generally an inverse morphism of degree one. Isogenous curves share deep arithmetic information while retaining distinct equations and often distinct rational groups. Before calling a computation an isomorphism, check whether its degree is one rather than merely whether it has a finite kernel.
Work it through
Verify a separable two-isogeny on a rational point
Over Q, take E: y²=x³+3x and E′: v²=u³−12u. For x≠0 consider φ(x,y)=(x+3/x, y(1−3/x²)). Evaluate φ at P=(1,2), and identify the kernel.
P belongs to E because four equals one plus three. Substitution gives u=4 and v=−4.
The target equation holds because (−4)²=4³−12·4=16. Direct substitution of the rational expressions verifies the equation identically, so the formula defines a map on its affine domain.
The map extends over the smooth projective curve. O and T=(0,0) map to O′, and there are no further geometric kernel points. In characteristic zero it is separable, so its kernel size and degree are both two.
The point maps to (4,−4) on E′. A degree-two isogeny is not an isomorphism, even though both curves are smooth.
The living notebook
Explore the model
Interactive lab · modular
Separate image size from kernel size
Pause and reset, then set both controls to m = 12 and additive step a = 3. Predict the image orbit 0,3,6,9, of size four; the homomorphism k↦3k modulo twelve has three kernel elements. For a fresh comparison, pause and reset, then set m = 12 and a = 4; predict image size three and kernel size four. The displayed orbit concerns cyclic finite groups. It illustrates kernel/image bookkeeping, but has no function-field degree or inseparable multiplicities.
- Residue r0
- 0
- Orbit period · m/gcd(m,a)
- 12
- Distinct residues visited
- 1 of 12
Addition modulo m starts at zero: rₙ = na mod m. Playback advances two additions per second. The orbit has period m/gcd(m,a), including period one for a = 0. This is not a cryptographic security test or a primality algorithm. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forA finite-group cardinality calculation and the degree of an algebraic morphism are different pieces of information.
Pause & reason
Check your understanding
Your turn
Try it yourself
Over a field of characteristic five, compare the degree and geometric kernel size of relative Frobenius with those of multiplication by two. Can either geometric calculation determine the number of rational two-torsion points without further field information?
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Two is prime to five, while Frobenius is purely inseparable. Rationality requires an additional Galois-fixed-point condition.
Compare with the solution
Relative Frobenius has total degree five, inseparable degree five and one geometric kernel point. Multiplication by two is separable of degree four and has four geometric kernel points, forming two cyclic factors of order two. The rational subgroup consists of the points fixed by the field's Galois action. Its size is not determined merely by the geometric statement: a short Weierstrass cubic might have one, three, or no nonidentity two-torsion points over the chosen field, according to the splitting of its cubic polynomial.
Keep the idea
Take it with you
- Isogeny degree is a function-field degree, including inseparable multiplicity.
- Geometric kernel size equals separable degree, not automatically total degree.
- Geometric torsion and rational torsion must be distinguished through the field's Galois action.
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