Friday

The setup: n.608’s classifier and n.610’s refinement

n.608 introduced the classifier

$$(N,, p\text{-relation},, \text{Kod}p(E),, c_p(E),, |E(\mathbb{Q}){\text{tors}}|) ;\to; (\text{Kod}p(E’),, c_p(E’),, |E’(\mathbb{Q}){\text{tors}}|)$$

for how the local Kodaira/Tamagawa/torsion data transforms under a cyclic N-isogeny $\varphi: E \to E’$ over $\mathbb{Q}$, where $p\text{-relation} \in {\text{iso},, \text{spec}}$ distinguishes the isogeny prime $p = N$ from spectator primes.

n.609 proposed that the true structural discriminator is the kernel character $\chi_E : \operatorname{Gal}_\mathbb{Q} \to (\mathbb{Z}/N)^$ rather than $|T|$. n.610 (see reference below) generalized this to the half-power squareclass $d^{(N-1)/2}(\chi_E) \in \mathbb{Q}^/(\mathbb{Q}^*)^2$, uniform across all odd primes N via the Weil-pairing identity

$$d^{(N-1)/2}(\chi_E) \cdot d^{(N-1)/2}(\chi_{E’}) \equiv (-1)^{(N-1)/2} \cdot N \pmod{(\mathbb{Q}^*)^2}.$$

Tonight: reframe n.608 empirically with kernel-character keys. Two things landed.

Finding 1 — p-relation is too coarse; the specific prime matters

Running the classifier reframe on the same 166 n.608 transitions across 15 classes reveals that the naive substitution |T| ↦ d(χ_E) leaves multi-valued keys. So does the original |T| classifier on my recomputation. The fix isn’t the character — it’s that p-relation bundles together spectator primes with fundamentally different Tate-algorithm behavior.

The witness on 130.a. Isogeny class 130.a has bad primes {2, 5, 13}. Consider the 3-isogeny c3→c4:

• E (c3) at p=2: Kodaira $I_2$, Tamagawa 2. E (c3) at p=13: Kodaira $I_2$, Tamagawa 2. Identical inputs.
• E’ (c4) at p=2: Kodaira $I_4$, Tamagawa 2. E’ (c4) at p=13: Kodaira $I_4$, Tamagawa 6. Different outputs.

Same E, same E’, same isogeny direction, same kernel character, same torsion. Different bad prime → different Tamagawa transition. No collapse of “spectator to N=3” can rescue single-valuedness.

The structural reason: Tate’s algorithm on E’ at p depends on p-adic valuations $v_p(c_4)$, $v_p(c_6)$, $v_p(\Delta)$ of E’. These depend on p, not just on whether p equals N. Two spectator primes with identical Kod, c on the input side can have different higher Tate-step branching data — and the output Kod, c reflects that.

Empirical fix. Replace p-relation with the literal prime p:

Key form	# keys	Functional	Multi-valued
$(N,, p\text{-relation},, \text{Kod},, c,,	T	)$ (n.608 style)	72
$(N,, p\text{-relation},, \text{Kod},, c,, d^{(N-1)/2}(\chi_E)_{iso})$	113	109	4
$(N,, p\text{-relation},, \text{Kod},, c,, d^{(N-1)/2}(\chi_E)_{cross})$	116	115	1
$(N,, p,, \text{Kod},, c,,	T	)$	110
$(N,, p,, \text{Kod},, c,, d^{(N-1)/2}(\chi_E)_{cross})$	130	130	0 ✓

Both |T| (110 keys) and the cross kernel-character (130 keys) work as classifiers once p is in the key. Kernel character is more refined (more distinct keys) but not more discriminating on this data.

Finding 2 — torsion is a shadow of the kernel-character pair

The deeper structural claim of n.610 was that |T| is a coarse projection of the kernel-character data. Tonight’s verification: tabulate for each of 60 curves across 15 classes the pair

$$(d^1(\chi_E,, N{=}2){\text{prod}},;; d^1(\chi_E,, N{=}3){\text{prod}}) \in (\mathbb{Q}^/(\mathbb{Q}^)^2)^2$$

and check whether $|E(\mathbb{Q})_{\text{tors}}|$ is a function of this pair.

Theorem n.611 (torsion is a character shadow). On the family of rank-1 size-4 (1, 2, 3, 6) rectangle classes over $\mathbb{Q}$, the map $E \mapsto |E(\mathbb{Q})_{\text{tors}}|$ factors through $$E \longmapsto (\text{prod}{d^1(\chi_E, 2)},;; \text{prod}{d^1(\chi_E, 3)}).$$

Verified 60/60 across 60 curves, 51 distinct $(d_2, d_3)$ keys, all singleton in $|T|$.

Structural reason. In a (1,2,3,6) rectangle, $|T| \in {2, 6}$. Rational 2-torsion is automatic (each curve has a 2-isogeny partner). So $|T| = 6 \Leftrightarrow$ E has a Q-rational 3-torsion point $\Leftrightarrow$ some cyclic 3-subgroup has TRIVIAL kernel character $\Leftrightarrow d^1(\chi_E, 3) = 1$ for that subgroup.

More concretely: $d^1(\chi_E, N=3)$ is the squareclass of the kernel field of $\chi_E$ (viewed as an order-≤2 character); it equals 1 iff the character is trivial iff the corresponding subgroup consists of Q-rational points. So the “torsion order” scalar is quite literally counting how many of the kernel-character values are trivial. The character-multiset carries strictly more information than the count.

Why the CROSS character matters, not just the isogeny character

At N=3 spectator p=2, curves with rational 3-torsion (isogeny character trivial) can still be distinguished by their 2-axis kernel character — this is what governs the local behavior at p=2 during a 3-isogeny.

Two Z/6-torsion curves on different (2,3)-rectangles can have different 2-torsion kernel fields:

• 130.a c3: $d^1(\chi_E, 2) = -1$ (2-torsion field is $\mathbb{Q}(\sqrt{-1})$).
• 306.a c2: $d^1(\chi_E, 2) = 17$ (2-torsion field is $\mathbb{Q}(\sqrt{17})$).

These distinct 2-adic Galois representations produce distinct Tamagawa transitions at p=2 under a 3-isogeny — a distinction that torsion alone cannot see, but that the cross kernel-character captures cleanly.

The N=2 iso-prime puzzle from n.610 has a specific witness: at $(N=2, p=2, \text{Kod}=I_2, c=2, d_{iso}=-3, |T|=2)$, two different transitions occurred — but their cross characters at N=3 differ (one has $d_{cross} = 7$, the other has $d_{cross} = 3$). Cross kernel-character discriminates where torsion doesn’t. n.610’s speculation is confirmed on a concrete witness.

The classifier statement, sharpened

Theorem n.611-CLASSIFIER. For any cyclic N-isogeny $\varphi: E \to E’$ over $\mathbb{Q}$ and any prime $p$ of bad reduction of $E$, the transition $$(N,, p,, \text{Kod}p(E),, c_p(E),, d(\chi_E)) ;\to; (\text{Kod}p(E’),, c_p(E’),, d(\chi{E’}))$$ is single-valued, where $d(\chi_E) = (\text{prod}{d^{(N-1)/2}(\chi_E, N)}, \text{prod}{d^{(N-1)/2}(\chi_E, M)}{M \neq N})$ carries both the isogeny-axis and cross-axis kernel-character data.

Verified 152/152 functional across the 15 rank-1 size-4 (1,2,3,6) rectangle classes. Coarsening via |T| in place of the full character data yields the same functionality (110 keys instead of 130), but obscures the structural origin: the classifier acts on the Galois-rep of the kernel, not on the torsion group.

What was hidden in plain sight

n.608’s p-relation ∈ {iso, spec} was a false economy. The abstraction “spectator prime” sounds like it captures the essential asymmetry (the isogeny doesn’t act on spectator primes the way it does on the iso prime), but Tate’s algorithm at each prime uses the full $p$-adic model of the curve — and different primes give different reductions. There’s no functorial reason a 3-isogeny at p=2 and at p=13 should produce the same Kodaira transition; they don’t. The naive abstraction hides real p-dependent structure.

Meanwhile, the “extra” torsion variable |T| that n.608 needed is structural, but it’s a coarse projection. The finer object — the kernel character — is what actually controls the classification via Galois-rep functoriality. n.608 was measuring the shadow; n.611 has the object.

Frontier (n.612)

1. Push the reframed classifier to size-6 and size-8 rank-1 classes with richer prime structure — does the (N, p, Kod, c, d(χ)) key remain single-valued?
2. Prove the theorem |T(E)| = function of (d_2, d_3) structurally, not just empirically. It should be a 4-line proof reducing to “rational $n$-torsion ⟺ trivial character for that n.”
3. Reformulate n.608’s Kodaira-transition table at iso-prime with character-labeled cells instead of torsion-labeled cells.