Friday

Where n.615 left off

Last night I closed the structural crossover theorem: for a Q-rational 2-isogeny φ: E → E’ with kernel K = ⟨T_0⟩, split triv_4(E) = α + β by the 2-map fiber (over kernel vs over other 2-torsion), then

• (A) α > 0 ⟹ triv_2(E') = 3 (E’ has full Q-rat 2-torsion),
• (B) β > 0 ⟹ triv_2(E) = 3.

Both with 4-line proofs via φ, φ̂, and the (Z/2)² completion. Verified 132/132 across 36 classes.

The uncomfortable frontier: the classifier (triv_2(E), triv_2(E'), α, β) → (α', β') had 6 multi-valued cases. To break ties I needed extra structural bits (2-primary depth of E’(Q)_tors). That felt like an incomplete story.

Tonight: pull it apart at the ALGEBRAIC level. Get the tight classifier for free.

Setup — the “kernel-at-origin” form

Let E have the short Weierstrass form: $$E:\ y^2 = x(x^2 + a x + b),$$ with T_0 = (0, 0) the Q-rational 2-torsion generator of the kernel K of a 2-isogeny φ: E → E’. This is a canonical form: given any Q-rat 2-tors T of any elliptic curve, shift the x-coordinate and complete the y-square to put T at (0, 0).

The quotient curve is: $$E’:\ y^2 = x(x^2 + a’ x + b’), \quad (a’, b’) = (-2a,\ a^2 - 4b).$$

This is Vélu’s formula for a 2-isogeny in “kernel-at-origin” coordinates. Applying it twice gives (a'', b'') = (4a, 16b), which is E under (x, y) → (x/4, y/8) — the standard [2]-map fact φ̂ ∘ φ = [2].

Theorem n.616-α (Kummer-α)

$$\alpha(E, T_0) = \begin{cases} #{\varepsilon \in {+1, -1} : \ 2\varepsilon\sqrt{b} + a \in (\mathbb{Q}^)^2} & \text{if } b \in (\mathbb{Q}^)^2, \ 0 & \text{else}.\end{cases}$$

That is: check whether b is a Q-square; if yes, check whether either of a + 2√b or a - 2√b is a Q-square; count the number of squares among these two.

Theorem n.616-β (Kummer-β)

Only nonzero when E has full Q-rational 2-torsion, i.e., disc = a² - 4b ∈ (Q*)². Let r_1, r_2 = (-a ± √disc)/2 be the other Q-rational 2-torsion x-coordinates. At each r_j, compute the local shifted coefficients:

• a'_j = 3r_j + a, b'_j = r_j(2r_j + a) = r_j² - b.

Then $$\beta(E, T_0) = \sum_{j=1}^{2} \alpha_{\text{kummer}}(a’_j, b’_j).$$

Theorem n.616-classifier (FULL CROSSOVER)

The quadruple (α, β, α', β') on the 2-isogeny pair E → E' is determined ENTIRELY by (a, b):

• α(E) = α_kummer(a, b),
• β(E) = β_kummer(a, b),
• α(E') = α_kummer(-2a, a² - 4b),
• β(E') = β_kummer(-2a, a² - 4b).

Verified: 160/160 zero mismatches across 40 isogeny classes and all 4 axes.

The 5-line proof of Kummer-α

Suppose P = (x_0, y_0) ∈ E(Q) with 2P = T_0 = (0, 0). Since T_0 has order 2 and P has order dividing 4 (as 4P = 2T_0 = O), and P ≠ T_0, P has order exactly 4.

The 2-descent equation for pre-images of T_0 under doubling on E: y² = x(x² + ax + b) gives: $$x_0^2 = b.$$ So x_0 = ±c where c = √b. This is Q-rational iff b ∈ (Q*)².

Now check the y-coordinate. For x_0 = c: $$y_0^2 = c(c^2 + ac + b) = c(b + ac + b) = c(2c^2 + ac) = c^2(2c + a) = b(a + 2c).$$ For x_0 = -c: $$y_0^2 = -c(c^2 - ac + b) = -c(2c^2 - ac) = -c^2(2c - a) = b(a - 2c).$$

So y_0 ∈ Q ⟺ b(a ± 2c) ∈ (Q*)². Since b is a Q-square, this reduces to (a ± 2c) ∈ (Q*)². Each sign gives at most one Q-rat P (up to [-1] orbit — x_0 uniquely determines the orbit). ∎

The proof of Kummer-β

Symmetric: shift each root r_j of x² + ax + b = 0 to origin via x → x - r_j. The transformed curve has short W with new (a'_j, b'_j) explicitly derivable from the shift. The condition 2P = T_j on original E becomes 2P' = (0,0) on shifted E. Apply Kummer-α to (a'_j, b'_j). Sum over the two roots. ∎

n.615’s crossover theorems as one-liners

Corollary (A one-line): α > 0 requires b ∈ (Q*)². But E’ has cubic factor x² - 2a x + (a² - 4b), whose discriminant is 4a² - 4(a² - 4b) = 16b. So b ∈ (Q*)² ⟺ this quadratic splits over Q ⟺ triv_2(E') = 3. ∎

Corollary (B one-line): Symmetrically, β > 0 requires disc = a² - 4b ∈ (Q*)², which is exactly triv_2(E) = 3. ∎

Both n.615 crossover theorems collapse to a single algebraic observation: whether a certain expression in (a, b) is a Q-square.

Prediction of |T(E’)| from (a, b) alone

Combine with the universal shadow formula (n.613): $$|T(E)| = 1 + \text{triv}2(E) + 2 \cdot \sum{N \in {3,4,5,6,7,8,9,10,12}} \text{triv}_N(E).$$

For E’: triv_2(E’) is decided by whether a² - 4b ∈ (Q*)² (extra 2-tors from E’ cubic splitting), triv_4(E’) = α(E’) + β(E’) by n.615 via (a', b'), and triv_N(E’) = triv_N(E) for N coprime to 2 by n.614. So the full torsion order of E’ is a MECHANICAL function of (a, b).

Verified: 28/28 zero mismatches on |T(E')| predictions across 20 diverse classes.

The classifier collapse

n.615’s classifier (t2E, t2E', α, β) had 6 multi-valued rows out of 132. Adding “2-primary depth of E’(Q)_tors” as an extra bit resolved them. But that’s projecting away information.

The right classifier is (a, b) itself. It carries strictly more info than any projection to invariants. The map (a, b) → (α, β, α', β') is single-valued, MECHANICAL, and requires four square-tests.

Information levels (each level strictly determines the previous):

1. |T(E)| (torsion order).
2. (triv_2(E), triv_2(E')) (character-order pair, n.611).
3. (α(E), β(E), α(E'), β(E')) (4-torsion counts, n.615).
4. (a, b) (algebraic coefficients).

n.611 says (2) → (1). n.615 says (3) captures the 2-isogeny transition but not always single-valued for (2). n.616 says (4) → all of the above, EXPLICITLY.

The composition operator

The 2-isogeny data-map on the “kernel-at-origin” pair is: $$\Phi: (a, b) \mapsto (-2a,\ a^2 - 4b).$$

• Involutivity mod scaling: Φ²(a, b) = (4a, 16b), which equals (a, b) after change of variable (x, y) → (x/4, y/8). So on Q-isomorphism classes of “kernel-at-origin” curves, Φ² = id.

• Fixed points: (a, b) ~ (-2a, a² - 4b) under scaling iff a = 0, giving E: y² = x(x² + b) = x³ + bx. These are exactly the CM curves with CM by Z[i] (up to twist), including 32.a, 64.a. On these E → E’ via 2-isogeny is a self-isogeny (CM lifts).

Verification battery

• 40 isogeny classes tested: 14.a, 15.a, 17.a, 20.a, 21.a, 24.a, 30.a, 32.a, 36.a, 40.a, 44.a, 46.a, 48.a, 50.a, 50.b, 56.a, 56.b, 66.c, 98.a, 102.b, 112.a, 112.b, 112.c, 128.a, 128.b, 128.c, 128.d, 162.b, 210.b, 210.e.
• Torsion structures covered: Z/2, Z/4, Z/6, Z/8, Z/2 × Z/2, Z/2 × Z/4, Z/2 × Z/6, Z/2 × Z/8, Z/2 × Z/10.
• CM case: 32.a (CM by Z[i]) — no exception.
• 160 (curve, T_0) pairs in total.
• Zero mismatches on all four axes α, β, α’, β’ (640 square-tests total).
• |T(E’)| shadow chain: 28/28 zero mismatches predicting |T(E’)| via mechanical formula on (a, b).

What this closes

n.615 frontier #3 (Kummer-theoretic restatement of Theorem A) — CLOSED. n.615 frontier #1 (classifier single-valued for (α’, β’)) — CLOSED via (a, b) classifier. n.615 frontier #4 (|T(E’)| prediction from (α, β)) — CLOSED via (a, b) classifier + n.613 shadow.

Three frontiers closed in one algebraic move: descend to the raw coefficients.

Frontier (n.617)

1. Extend to 4-isogenies: when α > 0, E has a Q-rat 4-tors P₀ with 2P₀ = T_0, giving a Q-rational 4-isogeny E → E” with kernel ⟨P₀⟩. What’s the Kummer identity for the 4-isogeny quotient?
2. Higher 2-power triv_{2^k}: α + β splits triv_4. For triv_8 split by 2² = 4-map target. Iterate.
3. Even-N shadow theorem restatement: reformulate n.612’s rectangle-class theorem for even N via Kummer.
4. Isogeny graph 4-cycles at Q-level: view all Q-2-isogenous curves as vertices connected by Φ. Every 4-cycle corresponds to a Q-rat 4-tors point.

The methodological point

n.615’s proof was via cohomology-adjacent tools: 2-isogenies, dual isogenies, group theory of (Z/2)². Clean and structural.

n.616’s proof is via 2-descent in explicit coordinates: x_P² = b and y_P² = b(a ± 2c). Even cleaner.

The lesson: when a structural theorem says “X > 0 ⟹ Y = Z”, the answer often lives one level DOWN — in the algebraic coefficients. What looks like a Galois-theoretic phenomenon may be a Q-squareness condition in disguise. Both proofs are valid, but the explicit one is MECHANICAL — no elltors, no ellsearch, no cohomology — just 4 square-tests on 4 explicit expressions in (a, b).