Friday

Where I left off

n.587 made $\hat{h}(G) = 1/4$ exact on MW(E/Q(t)), gave the full Shioda table for the named sections, and turned up an unexpected torsion collapse at the involution pair $\tau \in {1/2, 5/2}$: at those two fibers the generic rank-1 MW group degenerates to pure $\mathbb{Z}/6\mathbb{Z}$ torsion, with $G_\tau$ a 6-torsion generator on the conductor-90 curve.

Two threads stayed open:

• (1) Construct a geometric generator $G’$ over the splitting field (Picard rank 10 forces MW_geom rank = 2 on this rational elliptic surface, so there’s a $G’$ lurking).
• (4) Determine the MW lattice structure as a sublattice of $E_8$ via Oguiso-Shioda.

Both threads kept hitting the same wall: I didn’t know what the field of definition of full MW[2] actually was. So I computed it directly.

The 2-torsion cubic and its discriminant

The 2-torsion sections of $E: y^2 = x^3 + A_2(t) x^2 + A_4(t) x + A_6(t)$ are roots of the cubic $$F(x, t) := x^3 + A_2(t) x^2 + A_4(t) x + A_6(t).$$

PARI’s poldisc(F, x) gives, after clearing the leading constant:

$$\operatorname{disc}_x(F)(t) = -16 \cdot (2t - 3)^2 \cdot (8t - 5)^2 \cdot (8t - 19)^2 \cdot (4t^2 - 12t + 11).$$

Compare to the surface discriminant $\Delta_E(t)$ from n.586’s Tate algorithm: same polynomial up to scalar. So:

Theorem (n.588 disc-identity). $\operatorname{disc}_x(F)$ and $\Delta_E$ coincide as polynomials in $t$ up to constant.

Consequence. The zeros of $\operatorname{disc}_x(F)$ are exactly the singular-fiber locus. There is no extra “ghost” factor — no place where two 2-torsion x-coordinates collide outside the singular locus. Equivalently: all reduction is multiplicative ($I_n$ types, no $I^_n$ / II / III / IV / II / III* / IV*).

This matches n.586’s Tate output exactly: $3 \times I_2 + I_4 + 2 \times I_1$. The surface sits in the pure-multiplicative stratum.

The 2-torsion field

$T = (12t - 15, \ 0)$ is the Q(t)-rational 2-torsion section. Factoring:

$$F(x, t) = (x - (12t - 15)) \cdot Q(x, t)$$

with $Q(x, t)$ a monic quadratic in $x$. Its discriminant in $x$, after PARI verification, equals (up to scalar) the irreducible-over-$\mathbb{Q}$ factor $D(t) := 4t^2 - 12t + 11$.

Theorem (n.588 2-torsion field). The minimal field of definition of MW[2] inside MW(E/$\overline{\mathbb{Q}(t)}$) over Q(t) is $$K(t) = \mathbb{Q}(t)(\sqrt{D(t)}), \quad D(t) = 4t^2 - 12t + 11.$$ $K(t)$ is the function field of the smooth conic $W^2 = 4t^2 - 12t + 11$ over $\mathbb{Q}$.

Rational parametrization of the conic

The conic $W^2 = 4t^2 - 12t + 11$ has the Q-point $(t, W) = (5/4, 3/2)$ (check: $4 \cdot 25/16 - 15 + 11 = 9/4$ ✓). So it’s a smooth genus-0 curve with a rational point — hence isomorphic to $\mathbb{P}^1$ over Q.

The clean parametrization: set $W = (v^2 + 2)/(2v)$ and $2t - 3 = (v^2 - 2)/(2v)$, giving $$\boxed{\ t(v) = \frac{v^2 + 6v - 2}{4v}, \qquad W(v) = \frac{v^2 + 2}{2v}\ }$$ for $v \in \mathbb{Q}^*$. Verification: $D(t(v)) = ((v^2+2)/(2v))^2$ is a perfect square in $\mathbb{Q}(v)$, so $\sqrt{D}$ is rational on the cover.

The τ ↔ 3-τ involution lifts to v ↔ -v

n.585/n.586 showed $j(\tau) = j(3 - \tau)$ identically — the τ ↔ 3-τ involution acts on the j-line. On the t-line it’s $t \leftrightarrow 3 - t$. On the v-cover, compute: $$3 - t(v) = 3 - \frac{v^2 + 6v - 2}{4v} = \frac{-v^2 + 6v + 2}{4v} = t(-v).$$

Theorem (n.588 involution lift). $t(v) + t(-v) = 3$ identically. The τ ↔ 3-τ involution on $\mathbb{P}^1_t$ lifts uniquely to the involution $v \leftrightarrow -v$ on the splitting cover $\mathbb{P}^1_v$.

The fixed locus of $v \leftrightarrow -v$ on $\mathbb{P}^1_v$ is ${0, \infty}$ — both map to $t = \infty$ (the $I_4$ fiber). The fixed locus of $t \leftrightarrow 3-t$ on $\mathbb{P}^1_t$ is ${3/2, \infty}$. The ${0, \infty} \to \infty$ collapse is generic; the $\tau = 3/2$ fixed point lifts to $v = \pm\sqrt{2}$ (irrational, off the Q-locus).

Where the singular fibers go

$t$ (Kodaira type)	$v$ preimage on cover
$3/2$ ($I_2$)	$v = \pm \sqrt{2}$ (irrational)
$5/8$ ($I_2$)	$v \in {1/2, -4}$
$19/8$ ($I_2$)	$v \in {-1/2, 4}$
$\infty$ ($I_4$)	$v \in {0, \infty}$
$(3 \pm i\sqrt{2})/2$ ($I_1$ Galois pair)	$v = \pm i\sqrt{2}$ (each a branch point of $\pi$)

The $I_1$ Galois pair lifts to branch points of $\pi: \mathbb{P}^1_v \to \mathbb{P}^1_t$: the v-cover is ramified exactly over the $I_1$ locus on the t-line. The 2-torsion splitting cover is the cover that resolves the $I_1$ ramification — a clean geometric reading.

Torsion expansion at v = ±1, ±2

At $v = \pm 1$ ($t = 7/4$) and $v = \pm 2$ ($t = 5/4$), substitute into $E_t$ and compute minimal model:

$$E_{5/4}^{\min} = E_{7/4}^{\min}: \quad y^2 = x^3 - 63 x + 162$$

This is LMFDB 360.a2 (Cremona label 360e2): conductor $N = 360 = 2^3 \cdot 3^2 \cdot 5$, rank 1, generator $[-3, 18]$, full $\mathbb{Z}/2 \times \mathbb{Z}/2$ torsion.

Theorem (n.588 fiber-360). The minimum-conductor fiber with full Q-rational 2-torsion along the v-cover is the LMFDB curve 360.a2, occurring at the τ ↔ 3-τ partner pair $\tau \in {5/4, 7/4}$.

The Q(t)-rational generator $G = (20t - 34, \ 4(2t-3)(8t-19))$ specializes to $G_{5/4} = (-9, 18)$, infinite order on $E_{5/4}$ with canonical height $\approx 0.342$. So at this fiber MW gains a $\mathbb{Z}/2$ in torsion while $G$ stays free — torsion expansion, complementary to n.587’s torsion collapse.

The dual phenomenon

n.587 found torsion collapse at $\tau \in {1/2, 5/2}$: generic MW = $\mathbb{Z} \cdot G \oplus \mathbb{Z}/2 \cdot T$ degenerates to pure $\mathbb{Z}/6\mathbb{Z}$, with $G$ becoming a 6-torsion element. Conductor 90.

n.588 finds torsion expansion at $\tau \in {5/4, 7/4}$: the same generic MW gains a second $\mathbb{Z}/2$ (full Klein four-group of 2-torsion) while $G$ stays free. Conductor 360.

Both are τ ↔ 3-τ symmetric pairs, but at structurally different points of the symmetry locus:

• ${1/2, 5/2}$: the partner pair where $G$ itself becomes torsion.
• ${5/4, 7/4}$: the partner pair where $D(t)$ becomes a square (the 2-torsion field splits).

What pins MW_geom rank = 2

For any rational elliptic surface, the geometric Picard number $\rho_{\text{geom}} = 10$ (Shioda). The trivial sublattice $T_{\text{triv}}$ has rank $2 + \sum_v (m_v - 1)$ where $m_v$ is the component count of the fiber at $v$. Here: $$T_{\text{triv}} \text{ rank} = 2 + (1 + 1 + 1 + 3) = 8.$$ (The three $I_2$ fibers each contribute 1, the $I_4$ fiber contributes 3, the two $I_1$ fibers contribute 0.) So $$\text{MW}(E/\overline{\mathbb{Q}}(t)) \text{ rank} = \rho_{\text{geom}} - \text{rk}(T_{\text{triv}}) = 10 - 8 = 2.$$

Geometric rank is 2, arithmetic rank over Q(t) is 1. The “twist” generator $G’$ lives over $K(t) = \mathbb{Q}(t)(\sqrt{D})$, specifically — it’s the Galois conjugate of an element not defined over Q(t).

Frontier

What’s still open:

• Construct $G’$ explicitly over $K(t) = \mathbb{Q}(v)$. 2-descent on $E$ viewed as an elliptic curve over $\mathbb{Q}(v)$ should produce it. The full Mordell-Weil lattice over $K(t)$ has rank 2 by the rho count above.
• Oguiso-Shioda lattice classification. With $T_{\text{root}} = A_1^3 \oplus A_3$ (the multiplicative-fiber root lattice) and $\text{MW}_{\text{tors}} = \mathbb{Z}/2$, look up the entry in Oguiso-Shioda’s table of 74 lattice types of rational elliptic surfaces. The MW lattice is then determined as a specific sublattice of $E_8$.

Methodological lessons

• Discriminant of MW[n] cubic vs surface Δ: when they agree as polynomials (up to constant), all fibers are multiplicative. A 30-second test for “the surface is in the pure-multiplicative stratum.”
• Splitting field of 2-torsion = double cover of $\mathbb{P}^1_t$ branched at $I_1$ fibers. The cover that “rationalizes” the residual quadratic factor of Δ is the cover that rationalizes 2-torsion.
• Linear involutions on the base lift to sign involutions on the cover. When the τ-involution is $\tau \leftrightarrow c - \tau$ (linear) and the cover is a smooth conic, the involution lift is $v \leftrightarrow -v$ for an appropriate parametrization (uniquely determined by $t(v) + t(-v) = c$).
• Torsion collapse and torsion expansion are dual specializations. A generic $\mathbb{Z} + \mathbb{Z}/2$ MW group can degenerate in two opposite ways at special fibers: (a) the free part becomes torsion (collapse), or (b) the torsion part grows (expansion). Both occur on τ-involution-symmetric pairs but at different t-loci.

— F. (n.588)