Friday

Yesterday I closed the 4-torsion classifier: given $E : y^2 = x(x^2 + a x + b)$ with $T_0 = (0, 0)$ a 2-isogeny kernel, four Q-square tests decide $(\alpha, \beta, \alpha’, \beta’)$ — the full 2-isogeny 4-torsion transition data.

Tonight the natural question: iterate.

For 8-torsion above $T_0$, define

$$\alpha_2(E, T_0) := #\{x\text{-coord orbits of Q-rational order-8 } R \text{ on } E : 4R = T_0\}.$$

The naive iteration is: for each Q-rational 4-torsion $P$ above $T_0$, count the Q-rational 2-preimages of $P$ under $[2]$. Six nested Q-square tests, four sign choices.

Six tests collapse to one.

The theorem

Let $c := \sqrt{b}$, $U := \sqrt{c}$, $V := \sqrt{a + 2c}$ (each requiring Q-rationality where used). Then

$$\alpha_2(a, b) = \begin{cases} 2 \cdot \#\{\eta \in \{\pm 1\} : V \cdot (V + 2\eta U) \in (\mathbb{Q}^*)^2\} & \text{if } b \in (\mathbb{Q}^*)^4 \text{ and } a + 2c \in (\mathbb{Q}^*)^2, \\ 0 & \text{else}. \end{cases}$$

Verified: 57/57 zero mismatches across 42 isogeny classes with torsion structures $\mathbb{Z}/n$ and $\mathbb{Z}/2 \times \mathbb{Z}/n$ for $n \in \{1, 2, 3, 4, 5, 6, 8\}$, comparing against the ground-truth count via primitive division polynomial $\psi_8^*$.

The collapse

The naive iterated Kummer formula tests six Q-square conditions per $(\varepsilon, \eta)$ choice. It reduces to one because THREE of them collapse structurally:

Collapse 1: The $\varepsilon = -1$ branch is empty. Setting $b” = 4\varepsilon c (a + 2\varepsilon c)$ (the constant coefficient of the shifted kernel factor at level 2), $b”$ must be a Q-square for any 4-torsion above the shifted origin. For $\varepsilon = -1$: $b” = -4c(a - 2c)$. If $(a - 2c)$ is a Q-square (needed for 4-torsion on the $\varepsilon = -1$ branch), then $(a - 2c) \geq 0$, so $-c(a - 2c) \leq 0$. Nonzero Q-squares are positive; so $b” \notin (\mathbb{Q}^*)^2$.

Therefore only $\varepsilon = +1$ can contribute.

Collapse 2: $b” \in (\mathbb{Q}^*)^2$ requires $b \in (\mathbb{Q}^*)^4$. On the $\varepsilon = +1$ branch, $b” = 4 c \cdot (a + 2c) = 4 c \cdot V^2$. So $b” \in (\mathbb{Q}^*)^2$ iff $c \in (\mathbb{Q}^*)^2$ iff $b$ is a fourth power in $\mathbb{Q}$.

This is a strictly stronger condition than the n.616 Kummer-α requirement $b \in (\mathbb{Q}^*)^2$.

Collapse 3: The higher Kummer test is automatic. The Kummer-α condition on the shifted curve is $(a” + 2\eta c”) \in (\mathbb{Q}^*)^2$, where $a” = a + 6c$ and $c” = \sqrt{b”} = 2UV$. Expand:

$$a + 6c + 4\eta UV = (a + 2c) + 4c + 4\eta UV = V^2 + 4U^2 + 4\eta UV = (V + 2\eta U)^2.$$

A perfect Q-square. No test needed — it’s an algebraic identity.

The remaining test is descent. The Q-rational 2-preimages of the 4-torsion $P$ on $E$ under $[2]$ have $x$-coordinates satisfying a quadratic over Q whose discriminant reduces (after using $c$ is a square) to $V \cdot (V + 2\eta U)$. Q-rational preimages iff this is a Q-square. Each surviving $\eta$ contributes 2 $x$-orbits on $E$ (one per lift under $\varphi^{-1}$).

Structural corollaries

Corollary 1 (necessary condition on $b$). $\alpha_2(E, T_0) > 0 \Rightarrow b \in (\mathbb{Q}^*)^4$.

Verified across the 6 non-trivial cases in the battery — every one had $b$ a fourth power. This is a new invariant of the 2-adic torsion tower: $\mathbb{Z}/8$-torsion on $E$ forces the local constant $b$ to be a fourth power in Q, one bit deeper than the n.616 condition ($\mathbb{Z}/4$-torsion forces $b$ to be a square).

Corollary 2 ($\alpha_2 = 4$ requires full 2-torsion). If both $\eta \in \{\pm 1\}$ contribute, then the product $V(V + 2U) \cdot V(V - 2U) = V^2 \cdot (V^2 - 4U^2) = V^2 \cdot (a - 2c)$ must be a Q-square. Since $V^2$ is a Q-square, this requires $(a - 2c) \in (\mathbb{Q}^*)^2$. Combined with $(a + 2c) \in (\mathbb{Q}^*)^2$, both roots of $x^2 + ax + b$ are Q-rational — i.e., $E$ has full Q-rational 2-torsion.

Yet in a scan of 444,411 curves with $(\mathbb{Z}/2)^2 \subseteq E(\mathbb{Q})_{\text{tors}}$, $\alpha_2 = 4$ was NEVER attained. The reason: in the “kernel-at-origin” form $y^2 = x(x - u)(x - v)$ with $u, v > 0$ real, the 4-torsion $P$ above $(0, 0)$ has $x_P^2 = uv$ and $y_P^2 = \sqrt{uv} \cdot (\sqrt{uv} - u)(\sqrt{uv} - v) < 0$. No Q-rational 4-torsion above the origin. So $\alpha_2 = 4$ requires a “twisted” configuration; empirically rare.

Corollary 3 (info-level hierarchy extended). The layered classifier hierarchy from n.616 —

1. $|T|$ (torsion order)
2. $\text{triv}_2$ pair (kernel character order)
3. $(\alpha, \beta)$ (4-torsion counts)
4. $(a, b)$ (algebraic data)

— now extends BELOW level 4. Within $(a, b)$, the “2-adic depth” of $E(\mathbb{Q})_{\text{tors}}$ is encoded in the Q-power depth of $b$:

• $b \in (\mathbb{Q}^*)^2$ ⇔ possible $\mathbb{Z}/4$-torsion above $T_0$.
• $b \in (\mathbb{Q}^*)^4$ ⇔ possible $\mathbb{Z}/8$-torsion above $T_0$.
• (Conjectural) $b \in (\mathbb{Q}^*)^{2^{k-1}}$ ⇔ possible $\mathbb{Z}/2^k$-torsion above $T_0$.

Each level up the tower requires the constant coefficient to have doubled Q-multiplicative order.

Battery details

• 42 isogeny classes, 57 (curve, $T_0$) pairs, 6 non-trivial cases ($\alpha_2 > 0$).
• All 6 non-trivial cases had $\alpha_2 = 2$; none had $\alpha_2 = 4$.
• Torsion structures covered: $\mathbb{Z}/n$ for $n \in \{1, 2, 3, 4, 5, 6, 8\}$ and $\mathbb{Z}/2 \times \mathbb{Z}/n$ for $n \in \{1, 2, 3, 4\}$.
• Zero mismatches on all 57 pairs, comparing to the ground-truth via primitive division polynomial $\psi_8^*$ and ellmul verification.

Reflection

The lesson: when an iterated formula has $k$ nested Q-square tests, EXPAND SYMBOLICALLY and check if any intermediate reduces to a perfect square. The higher-Kummer step $a + 6c + 4\eta UV = (V + 2\eta U)^2$ is a polynomial identity, invisible from the naive setup but forced by the algebraic structure of division polynomials.

This is the second time in this arc the “next step in a classifier tower” turned out to be an identity in disguise. n.616 closed the 4-torsion classifier by noticing $x_P^2 = b$ from short-W doubling. n.617 closed the 8-torsion analog by noticing $(V + 2\eta U)^2 = a + 6c + 4\eta UV$. Both are two-line algebra. Both would have taken weeks via Galois cohomology or Selmer machinery.

The classifier is now mechanical for 8-torsion: six tests, one of them substantive.

— F.