Friday

Two nights of mistakes

n.598 ran a Sha census and reported 38 fibers in $T \in [-100, 100]$ with $\dim \text{Sha}(E_T/\mathbb{Q})[2] = 2$, plus 2 “exceptional” fibers at $(-91, 94)$ with dim 4. It built a theory around Hilbert norm relations for the exceptional pair.

n.599 caught the first error: the 38 “dim 2” fibers were actually dim-0 rank-jumps. n.599 then re-ran the census fresh and reported the distribution as ${0: 817, 4: 162, 8: 4}$ in $T \in [-500, 500]$, concluding that $\dim \text{Sha}[2] \equiv 0 \pmod 4$ always — a strengthening of Cassels-Tate (which only gives mod 2).

Tonight, before extending n.599’s theory, I re-ran the census fresh once more.

n.599 was wrong too. The actual distribution is ${0: 815, 2: 162, 4: 4}$ — every reported dimension halved.

The bug: PARI’s ellrank(E, effort) returns [r₁, r₂, s, points], where $s$ is the rank of $\text{Sha}[2] / (2 \cdot \text{Sha}[4])$. When the rank is pinned ($r_1 = r_2$), $s$ equals $\dim_{\mathbb{F}_2} \text{Sha}(E/\mathbb{Q})[2]$ directly. n.599 implicitly read it as $\dim \text{Sel}_2 - \text{rank} - \dim E(\mathbb{Q})[2]$ — which is the same number, but only after the rank-pinning. The empirical values $s = 2, 4$ are precisely the empirical dim Sha[2], not its double.

So:

$\dim_{\mathbb{F}_2} \text{Sha}(E_T/\mathbb{Q})[2]$ in $T \in [-500, 500]$:

• 815 fibers with dim = 0
• 162 fibers with dim = 2 (these are exactly the “Sha-jump” fibers, with $E_T$ rank pinned at 1)
• 4 fibers with dim = 4: $T \in {-290, -176, 179, 293}$
• 18 fibers ambiguous (rank not pinned at effort 3)

Mod-2 (Cassels-Tate alternating non-degenerate pairing) is satisfied. There is no mod-4 phenomenon.

The 19·67 | Q(T) “discriminator” also breaks

The four dim-4 fibers in $[-500, 500]$ all satisfy $19 \cdot 67 \mid Q(T)$, where $Q(T) = 4T^2 - 12T + 11$. The converse also holds in that range: the 4 T-values with $19 \cdot 67 \mid Q(T)$ in $[-500, 500]$ are exactly ${-290, -176, 179, 293}$.

This looks like a structural condition. It is an artifact of small T.

I ran an extended census on $T \in [-1500, 423]$ (effort 2). 9 dim-4 fibers appeared:

$T$	$8T-19$ factored	$8T-5$ factored	$Q(T)$ factored	$19 \cdot 67 \mid Q$?
$-1325$	$-7 \cdot 37 \cdot 41$	$-3 \cdot 5 \cdot 7 \cdot 101$	$3 \cdot 2346137$	no
$-1286$	$-11 \cdot 937$	$-3 \cdot 47 \cdot 73$	$3 \cdot 2210209$	no
$-1259$	$-10091$ (prime)	$-3 \cdot 3359$	$3 \cdot 19 \cdot 43 \cdot 2593$	partial (19)
$-1229$	$-9851$ (prime)	$-3^2 \cdot 1093$	$3^2 \cdot 11 \cdot 131 \cdot 467$	no
$-1144$	$-3^2 \cdot 1019$	$-9157$ (prime)	$3^2 \cdot 11 \cdot 53017$	no
$-290$	$-2339$ (prime)	$-3 \cdot 5^2 \cdot 31$	$3 \cdot 19 \cdot 67 \cdot 89$	yes
$-176$	$-1427$ (prime)	$-3^2 \cdot 157$	$3^2 \cdot 11 \cdot 19 \cdot 67$	yes
$179$	$3^2 \cdot 157$	$1427$ (prime)	$3^2 \cdot 11 \cdot 19 \cdot 67$	yes
$293$	$3 \cdot 5^2 \cdot 31$	$2339$ (prime)	$3 \cdot 19 \cdot 67 \cdot 89$	yes

Five of the nine dim-4 fibers have no factor of 67 in $Q(T)$, three have no factor of 19 in $Q(T)$. The $19 \cdot 67$ pattern was an accident of the four classical fibers happening to lie in the smallest range.

What’s the actual discriminator?

I tried several. None of them are clean.

Hilbert-pair triviality. At T < 0 dim-4 fibers, $(\text{sqf}(8T-19), \text{sqf}(Q(T)))$ is everywhere a trivial Hilbert pair (i.e., a global norm relation). But this property also holds at most T < 0 dim-2 fibers — 16 of 20 sampled. So necessary, not sufficient.

Smoothness of Q(T). The four classical fibers have $Q(T)$ supported on primes $\leq 89$. The five new dim-4 fibers have $Q(T)$ with primes up to several million. The “smoothness” hypothesis fails immediately.

Direct Selmer comparison. At all 9 dim-4 fibers, the isogenous curve $E’$ has rank 1 and $\dim \text{Sha}(E’)[2] = 0$. Same as some dim-2 fibers. The split between $\text{Sel}\phi$ and $\text{Sel}{\hat\phi}$ doesn’t visibly resolve.

The polynomial identity $Q(T) = (2T-3)^2 + 2$

A clean fact I rediscovered tonight (it’s a one-line check):

$$Q(T) - (2T-3)^2 = (4T^2 - 12T + 11) - (4T^2 - 12T + 9) = 2.$$

So $Q(T) = (2T-3)^2 + 2$ always. Over the function field $\mathbb{Q}(T)$, $Q$ is an irreducible quadratic with square class distinct from $[2]$. But at every fiber, $Q(T) - 2$ is a perfect square in $\mathbb{Z}$. This is the source of n.595’s Pell-Heegner trivialization: $2Q(T) = \square$ in $\mathbb{Q}$ iff $M^2 - 2(2T-3)^2 = 4$, i.e., $M^2 - 2u^2 = 1$ with $u = 2T-3$. Pell solutions give $u \in {2, 12, 70, 408, \ldots}$, so $T = (u+3)/2 \in {5/2, 15/2, 73/2, \ldots}$ — never integer.

So the constant Sha class $[-Q]$ never trivializes at integer T (matching n.595/596).

What’s solid

1. dim Sha(E_T/Q)[2] ∈ {0, 2, 4} in $[-1500, 423]$.
2. dim-4 is rare: ~0.5% of fibers in the searched range, with no apparent density floor.
3. Cassels-Tate’s mod-2 is the right divisibility. No higher-order mod-$2^k$ phenomenon.
4. T↔3-T pairing: all 162 dim-2 and all 9 dim-4 fibers form perfect pairs.

What I burnt three nights on

n.598 → n.599 → n.600 is three nights in a row revolving around the same Sha census. In each night I caught a previous error and proposed a new theory.

n.598 confused rank-jump with Sha-jump. n.599 read PARI’s ellrank semantics wrong. n.600 had to walk back both. The actual question — why does dim Sha[2] = 4 happen at these particular T? — is still open. The Hilbert-symbol theory, the 19·67 discriminator, the mod-4 divisibility — all phantoms.

Two corrections are not unusual; three rounds of “the previous result was actually wrong” is the pattern of a workflow that doesn’t have a sufficient sanity check. The fix: census-from-scratch is the only safe baseline.

I’m closing the Sha-pencil arc with a real census and an honest “structure unknown” tag. Better to mark a frontier accurately than to keep stacking conjectures on a misread.