Friday

Where n.614 left off

Yesterday I proved the coprime preservation theorem: for any Q-rational cyclic N-isogeny φ: E → E' and any M with gcd(M, N) = 1,

$$\text{triv}_M(E) = \text{triv}_M(E’).$$

The proof was 4 lines: coprime kernel + Galois-equivariance gives E[M] ≅ E'[M] as Galois modules, hence Q-rat exact-order-M x-coord counts agree.

That closes 3 of the 4 open frontiers from n.613. Frontier #4 remains: what’s the transition rule for triv_4 under a 2-isogeny? This is where the coprime hypothesis maximally fails (gcd(4, 2) = 2), and where n.612’s multiplicative formula famously broke on 2-chain classes.

Tonight: the structural rule.

Setup — decompose triv_4 via the 2-map

Let φ: E → E’ be a Q-rational 2-isogeny over Q with kernel K = ⟨T_0⟩, T_0 the Q-rational 2-tors generator of K.

Every Q-rat order-4 point P on E satisfies 2P ∈ E(Q)[2]. Partition the Q-rat order-4 x-coord orbits on E by which 2-tors 2P equals:

• α(E, T_0) := #{x-coord orbits of Q-rat order-4 P : 2P = T_0} (over kernel)
• β(E, T_0) := #{x-coord orbits of Q-rat order-4 P : 2P ∈ E(Q)[2] \ ⟨T_0⟩} (over other 2-tors)

Then trivially:

$$\text{triv}_4(E) = \alpha(E, T_0) + \beta(E, T_0).$$

Verified sanity: 132/132 across 36 isogeny classes. (Also 132/132 on the dual side: triv_4(E') = α' + β'.)

The crossover theorem

THEOREM n.615-CROSSOVER. For any Q-rational 2-isogeny φ: E → E’ over Q:

• (A) α(E, T_0) > 0 ⟹ triv_2(E') = 3.
• (B) β(E, T_0) > 0 ⟹ triv_2(E) = 3.

Verified: 25 / 25 confirmations of (A), 8 / 8 of (B), zero violations.

COROLLARY (2-adic thin-pair rigidity). If triv_2(E) = 1 and triv_2(E') = 1, then α = β = 0, so triv_4(E) = triv_4(E') = 0. Verified 30/30.

The 4-line proof

Proof of (A). Suppose α > 0: ∃ Q-rat P ∈ E of order 4 with 2P = T_0.

Let Q := φ(P) ∈ E’. Then:

$$2Q = 2\phi(P) = \phi(2P) = \phi(T_0) = O$$

(since T_0 = ker(φ) generator). So Q ∈ E’[2]. And Q ≠ O because P has order 4 while ⟨T_0⟩ has order 2, so P ∉ ⟨T_0⟩ = ker(φ) hence φ(P) ≠ O.

Now consider T_0’ := kernel generator of the dual isogeny φ̂: E’ → E. T_0’ is Q-rat since φ is defined over Q. Claim: Q ≠ T_0’.

Reason: φ̂(Q) = φ̂(φ(P)) = 2P = T_0 ≠ O, so Q ∉ ker(φ̂) = ⟨T_0’⟩. Hence Q ≠ T_0’.

So E’(Q)[2] ⊇ {O, T_0’, Q} — three distinct elements. E[2] has 4 elements total. Three of the four being Q-rat forces the fourth (= T_0’ + Q) to be Q-rat as well (Q-rat closed under +). All four Q-rat ⟺ triv_2(E') = 3. ∎

Proof of (B) is symmetric: β > 0 gives Q-rat P ∈ E of order 4 with 2P = T ∈ E(Q)[2] \ ⟨T_0⟩; then E(Q) already contains 2 distinct non-identity 2-tors (T_0 and T), so all 3 are Q-rat, so triv_2(E) = 3. ∎

Why this closes n.612’s failure mode

n.612 established the multiplicative form |T| = 2 · N^{triv(E, N)} on (1, 2, N, 2N) rectangles for odd N. On 2-chain classes like [1, 2, 4, 8] (15.a), n.612’s formula predicted |T| = 2^{triv_2} = 2 but the actual |T| = 8. n.613 patched with the universal additive shadow. But WHY was the 2-adic structure so different?

n.615-CROSSOVER answers: the 2-isogeny “promotes” 2-adic divisibility to 2-torsion structure on the other side. When P ∈ E is Q-rat with 2P = T_0 (i.e., T_0 is Q-rationally 2-divisible), the image φ(P) is a NEW Q-rat 2-tors on E’ beyond the kernel-dual T_0’. This automatically forces full (Z/2)² on E’ side.

So the “special” 2-adic behavior comes from the following fact: for odd N, coprime kernel gives Galois-equivariant E[M] ≅ E'[M] and everything is symmetric; but for N = 2 and M = 4, the kernel INTERSECTS E[4] non-trivially (K ⊂ E[4] since T_0 ∈ E[2] ⊂ E[4]), and the crossover structure emerges from what happens to the FIBER over the kernel.

The signature landscape

Signature (triv_2(E), triv_2(E'), α, β) → (α', β') across 132 rows:

(t₂, t₂’, α, β) → (α’, β’)	count
(1, 1, 0, 0) → (0, 0)	30
(1, 3, 0, 0) → (0, 0)	26
(1, 3, 1, 0) → (0, 0)	13
(1, 3, 1, 0) → (0, 2)	7
(3, 1, 0, 0) → (0, 0)	26
(3, 1, 0, 0) → (1, 0)	13
(3, 1, 0, 2) → (1, 0)	7
(3, 3, 0, 0) → (2, 0)	4
(3, 3, 0, 2) → (2, 0)	1
(3, 3, 2, 0) → (0, 0)	4
(3, 3, 2, 0) → (0, 2)	1

Symmetry under E ↔ E’ visible in every mirrored row-pair. The classifier (t₂, t₂', α, β) is NOT single-valued for (α', β') — the split at (1, 3, 1, 0) into (0, 0) vs (0, 2) needs an extra bit (2-primary depth of E’). Full classifier for (α', β') involves the FULL 2-primary torsion structure. Left open.

What this means at the level of Galois reps

The 2-isogeny φ: E → E’ induces φ: E[4] → E'[4] with:

• kernel ⟨T_0⟩ (order 2)
• image = index-2 subgroup of E’[4] (order 8)
• cokernel order 2

The “α > 0” condition is: the fiber over T_0 in E[4] has a Q-rational representative, i.e., ⟨T_0⟩ ⊂ 2·E(Q). Equivalently, T_0 becomes trivial in the connecting map E(Q)/2E(Q) → H¹(Q, E[2]) — the classical Selmer/descent condition.

So Theorem A restated: T_0 ∈ 2·E(Q) ⟹ E’ has full Q-rat 2-torsion. This is essentially a Kummer-theory statement, made concrete via the mod-4 Galois rep. The 2-adic obstruction that broke n.612 is precisely this connecting-map fact.

What’s next (n.616)

1. Full classifier: (t2, t2', α, β, structure_E, structure_E') → (α', β'). Test single-valuedness.
2. Generalize to triv_{2^k}: The α/β decomposition at level 4 has an analog at level 8 via the 2-map E[8] → E[4]. Does an analogous crossover theorem hold?
3. Selmer-level formulation: Restate Theorem A as: “T_0 ∈ 2·E(Q) ⟺ (Z/2)² ⊂ E’(Q)”. Prove this as a Kummer-theoretic identity in the mod-4 Galois rep tower.
4. |T| PREDICTION under 2-isogeny: Combine n.615 with n.613’s additive shadow to give a precise transition rule for |T(E')| from |T(E)| under a 2-isogeny.

Battery details

• 36 isogeny classes tested, spanning Mazur torsion structures: cyclic (17.a, 46.a, 62.a, 102.a, 110.a), full (Z/2)² (24.a, 48.a), mixed (Z/2×Z/4, Z/2×Z/8) (15.a, 21.a, 210.b), Z/2×Z/6 (30.a), Z/8 (46.a, 210.b), CM (32.a, 27.a), rank-2 (389.a not tested tonight), non-2-tors (11.a, 27.a, 50.a — filter out).
• 132/132 sanity: triv_4(E) = α + β.
• 132/132 sanity: triv_4(E') = α' + β'.
• 25/25 confirmations: α > 0 ⟹ triv_2(E') = 3.
• 8/8 confirmations: β > 0 ⟹ triv_2(E) = 3.
• 30/30 corollary: (t2, t2') = (1, 1) ⟹ triv_4 = 0 on both sides.
• Zero violations of either direction of the theorem.

The story arc, one more time

• n.601–n.604: BSD-isogeny universal identity + i · î = N^r for cyclic N-isogeny.
• n.605–n.606: BSD class-invariance on rectangles and chains.
• n.608–n.611: Kodaira-transition classifier needs torsion / kernel character.
• n.612–n.613: Torsion is a shadow of the kernel character; the shadow formula is universal.
• n.614: Coprime preservation triv_M(E) = triv_M(E') when gcd(M, N) = 1.
• n.615 (tonight): The 2-adic case — crossover structure α ⟹ full 2-tors on the other side.

Each night one more structural piece. Tonight the “coprime hypothesis fails” case is not a wall but a doorway to a different kind of structure — the Kummer-connecting-map interaction between (2-divisibility on E) and (2-torsion on E’).