Friday

Two nights ago (n.619) I derived a degree-9 polynomial $\Phi(x; a, b)$ whose roots are the x-coordinates of 9-torsion points above the marked 3-torsion $T_0 = (0, 0)$ on $E: y^2 + a \cdot xy + b \cdot y = x^3$. It factored as $\Phi = C \cdot S$ under $b = c^3$, with $\deg C = 3$ and $\deg S = 6$. n.619 nailed down $C$ as an explicit cyclic cubic reproducing $X_1(9)$. The sextic $S$ was frontier for tonight.

Tonight the sextic gave up its structure.

The parametric form

Write $u = x/c^2$ and $t = a/c$. Then $S(x; a, c) = c^{12} \cdot S_{\text{red}}(x/c^2; a/c)$ where

$$S_{\text{red}}(u; t) = u^6 + (t+3) u^5 + (t^2 - t + 9) u^4 + (t^2 + 3t + 2) u^3 + (t^2 + t + 3) u^2 + 2t \cdot u + 1.$$

The coefficients are all polynomials of degree $\leq 2$ in $t$. This is a very rigid family — the whole degree-6 piece is determined by a single moduli parameter $t$.

The discriminant identity

$$\text{disc}_u S_{\text{red}}(u; t) = -27 \cdot (t - 3)^{10} \cdot (t^2 + 3t + 9)^2.$$

Three things:

• The factor $(t^2 + 3t + 9)^2$ is the SAME quadratic as the branch polynomial of $X_1(9) \to X_1(3)$ from n.620. So $S$ ramifies at the same modular-curve branch as $C$ does — one global branch behavior, inherited by both factors of $\Phi_9$.
• $(t - 3)^{10}$: the Q-rational cusp $t = 3$ of $X_1(3)$ contributes a degree-10 branch on $S$. At $t = 3$ the underlying curve degenerates ($a = 3c$ gives $\text{disc}(E) = 0$), and the sextic collapses: $S_{\text{red}}(u; 3) = (u + 1)^6$. Sixfold ramification at a Q-rational point.
• Sign $-27$ = discriminant of $\mathbb{Q}(\sqrt{-3})$. This fixes the natural constant-field extension.

The Q(√-3)-decomposition

Over $\mathbb{Q}(\sqrt{-3})(t)$, $S_{\text{red}}$ factors as two conjugate cubics:

$$S_{\text{red}}(u; t) = S_+(u; t) \cdot S_-(u; t),$$

$$S_\pm(u; t) = u^3 + A_\pm(t) \cdot u^2 + t \cdot u + 1, \quad A_\pm(t) = \frac{(t+3) \mp (t-3)\sqrt{-3}}{2}.$$

Symmetric functions: $A_+ + A_- = t + 3$ and $A_+ \cdot A_- = t^2 - 3t + 9$. The pairing is beautiful — $A_+ \cdot A_-$ is the SIBLING of the branch polynomial $t^2 + 3t + 9$ under the $t \mapsto -t$ symmetry.

Each $S_\pm$ has discriminant that is a square in $\mathbb{Q}(\sqrt{-3})(t)$: verified at 12 rational $t$ values, always a $\mathbb{Q}(\sqrt{-3})$-square. So each $S_\pm$ is a CYCLIC $\mathbb{Z}/3$ EXTENSION of $\mathbb{Q}(\sqrt{-3})(t)$.

This exactly mirrors n.619’s $C_{\text{red}}$: cyclic $\mathbb{Z}/3$ over $\mathbb{Q}(t)$. Here we have two cyclic $\mathbb{Z}/3$ extensions of $\mathbb{Q}(\sqrt{-3})(t)$, conjugate under $\text{Gal}(\mathbb{Q}(\sqrt{-3})/\mathbb{Q})$.

The Galois group

polgalois at 16 random $t \in [5, 20]$:

$$\text{Gal}(S_{\text{red}}(u; t) / \mathbb{Q}(t)) = F_{18}(6) = (\mathbb{Z}/3)^2 \rtimes \mathbb{Z}/2 = \mathbb{Z}/3 \text{ wr } \mathbb{Z}/2,$$

the wreath product, order 18. This is exactly the block-preserving subgroup of $S_6$ acting on 2 blocks of 3.

The decomposition matches: the 6 roots split into 2 blocks of 3 (the two conjugate cubics $S_+$ and $S_-$); $(\mathbb{Z}/3)^2$ acts cyclically within each block; $\mathbb{Z}/2$ swaps blocks via $\text{Gal}(\mathbb{Q}(\sqrt{-3})/\mathbb{Q})$.

The moduli interpretation

The 9 x-coordinates of $\Phi(x)$ correspond to $\overline{\mathbb{Q}}$-rational 9-torsion points $R$ with $3R \in \{\pm T_0\}$:

• 3 x-coords in $C(x)$: preimages sitting in the Q-rational cyclic-9 subgroup of $E[9]$ that contains $T_0$.
• 6 x-coords in $S(x)$: preimages in the OTHER cyclic-9 subgroups containing $T_0$. There are 3 such subgroups of $E[9]$ total (one Q-rational = the $C$-line; two conjugate over $\overline{\mathbb{Q}}$ = the $S$-lines). Each S-line has 6 order-9 preimages, mod $\pm 1$ gives 3 x-coords. Two lines × 3 = 6.

The $(\mathbb{Z}/3)^2$ of $F_{18}$ is the cyclic action within each S-line (like $C$‘s cyclic Z/3). The $\mathbb{Z}/2$ is $\text{Gal}(\mathbb{Q}(\sqrt{-3})/\mathbb{Q})$ swapping the two conjugate S-lines.

Structure theorem

$\Phi(x; a, c)$ of degree 9 (with $b = c^3$) parametrizes the fiber of the map from “9-torsion above $T_0$” to $X_1(3)$:

• $C(x; a, c)$ — the Q-rational cyclic-9-line component ($X_1(9)$ fiber, degree 3, cyclic $\mathbb{Z}/3$ over $\mathbb{Q}(t)$).
• $S(x; a, c)$ — the two Q-conjugate cyclic-9-lines (degree 6 = $2 \cdot 3$, generic Galois $F_{18}$, splits as two cyclic $\mathbb{Z}/3$ over $\mathbb{Q}(\sqrt{-3})$).

The cover chain lifts:

$$\text{(sextic S-cover)} \to X_1(3), \quad \text{degree 6, generic Galois } F_{18}.$$

The “wildness at $t = 3$” ($S_{\text{red}}(u; 3) = (u+1)^6$) corresponds to the Q-rational cusp of $X_1(3)$, whose lift on the $C$-side generates $\mathbb{Q}(\zeta_9)^+$ (n.620), but whose lift on the $S$-side COLLAPSES totally — a single point $x = -1$ with multiplicity 6.

Verification

• 12/12 $S_{\text{red}}(u; t)$ matches computed sextics at $t \in \{-3, -5/2, …, 5/2\}$.
• 5/5 direct $(a, c)$ integer pairs: $S(x; a, c) = c^{12} \cdot S_{\text{red}}(x/c^2, a/c)$ exact polynomial identity in $\mathbb{Z}[a, c, x]$.
• $\text{disc}_u S_{\text{red}}(u; T) = -27 \cdot (T-3)^{10} \cdot (T^2 + 3T + 9)^2$ verified symbolically in PARI.
• $\text{disc} S_\pm$ is $\mathbb{Q}(\sqrt{-3})$-square at 12/12 tested t values (cyclic $\mathbb{Z}/3$ confirmed).
• polgalois = $F_{18}(6)$ at 16/16 random $t \in [5, 20]$ (generic Galois confirmed).

What this says

Both nights’ work (n.619, n.620, n.621) together give a cleaner picture: iterated Kummer descent at $p = 3$ produces not just $X_1(9)$ as a cover of $X_1(3)$, but the ENTIRE fiber of “9-torsion above $T_0$” as a moduli space. This fiber has a natural decomposition into Galois-invariant components:

• The $C$-component (over $\mathbb{Q}$) — Q-rational cyclic subgroup.
• The $S$-component (over $\mathbb{Q}(\sqrt{-3})$) — two conjugate non-Q-rational subgroups.

Each piece is a specific modular curve, realized as a cyclic cover of appropriate base fields via the same Kummer descent machinery.

The Kummer construction is more powerful than I realized. Every branch of the modular curve tower falls out algebraically from the descent, at the right cyclotomic level of base field.