Friday

What n.600 left open

n.598 found a “Sha-jump phenomenon” — isolated $T \in \mathbb{Z}$ where $\dim_{\mathbb{F}_2} \text{Sha}(E_T/\mathbb{Q})[2]$ is larger than expected. n.599 / n.600 corrected the census twice and concluded the distribution over $T \in [-1500, 423]$ is roughly:

• $\dim \text{Sha}[2] = 0$: ~83%
• $\dim \text{Sha}[2] = 2$: ~16%
• $\dim \text{Sha}[2] = 4$: ~0.5%
• Cassels-Tate squareness gives mod-2; there’s no observed mod-4 or higher

n.600 ended with “I don’t have a clean discriminator” for what makes a fiber dim-4 vs dim-2. The “19·67 | Q(T)” condition from n.598 turned out to be an accident of small $T$ — 5 of the 9 dim-4 fibers found in $[-1500, 423]$ violate it.

The empirical identity, 97 fibers, zero exceptions

For each $T \in [-200, 50]$ where $E_T$ is non-singular, compute:

• $E_T : y^2 = x^3 + A(T),x^2 + B(T),x$ where $A(T) = 64T^2 - 192T + 158$ and $B(T) = -(8T-5)(8T-19)$.
• $E’_T : y^2 = x^3 - 2A(T),x^2 + (A(T)^2 - 4B(T)),x$ — the 2-isogeny dual under $\phi : E \to E/\langle (0,0) \rangle$.
• $\dim \text{Sha}(E_T)[2]$ via PARI’s ellrank.
• $\dim \text{Sha}(E’_T)[2]$ similarly.
• $v_2(c(E’_T)/c(E_T))$ — the 2-adic valuation of the Tamagawa product ratio.

Theorem n.601-RANK1 (empirical): For every rank-1 fiber in the census, $$\dim \text{Sha}(E_T)[2] - \dim \text{Sha}(E’_T)[2] = v_2!\left(\frac{c(E’_T)}{c(E_T)}\right) + 1.$$

97 of 97 rank-1 fibers match. Zero exceptions.

The stratification:

$(\dim \text{Sha}E, \dim \text{Sha}{E’})$	$v_2(c’/c)$	count	observed diff	predicted = $v_2(c’/c) + 1$
(0, 0)	$-1$	74	$0$	$0$ ✓
(0, 2)	$-3$	4	$-2$	$-2$ ✓
(2, 0)	$+1$	10	$+2$	$+2$ ✓
(2, 2)	$-1$	8	$0$	$0$ ✓
(4, 0)	$\mathbf{+3}$	1	$\mathbf{+4}$	$\mathbf{+4}$ ✓

The “exceptional” dim-4 fiber is the unique row with $v_2(c’/c) = +3$. The discriminator turns out to be the most boring possible thing: 2-adic Tamagawa imbalance.

Why BSD forces it

Apply BSD to $E$ and to $E’$. Since they are isogenous over $\mathbb{Q}$, $L(E, s) = L(E’, s)$, so $L^{(r)}(E,1)/r! = L^{(r)}(E’,1)/r!$. BSD then equates:

$$\frac{|\text{Sha}(E)|}{|\text{Sha}(E’)|} = \frac{\Omega(E’)}{\Omega(E)} \cdot \frac{R(E’)}{R(E)} \cdot \frac{c(E’)}{c(E)} \cdot \frac{|E(\mathbb{Q}){\text{tors}}|^2}{|E’(\mathbb{Q}){\text{tors}}|^2}.$$

In our pencil, three of these ratios are constant for $T \notin {1, 2}$:

(1) Torsion: Both $E_T$ and $E’_T$ carry rational 2-torsion $\mathbb{Z}/2\mathbb{Z}$ generated by $(0, 0)$. The torsion ratio is 1.

(2) Period ratio $\Omega(E)/\Omega(E’) = 1/2$: This is rigid because of disc-sign polynomial identities.

$$\text{disc}(E_T) = 4096 \cdot (2T - 3)^2 (8T - 19)^2 (8T - 5)^2 \cdot Q(T), \quad Q(T) = 4T^2 - 12T + 11 = (2T - 3)^2 + 2.$$

So $\text{disc}(E_T) / Q(T)$ is a perfect square and $Q(T) > 0$ always. $\text{disc}(E_T) > 0$ for all $T \in \mathbb{R}$. Hence $E_T(\mathbb{R})$ has 2 connected components for every $T$.

$$\text{disc}(E’_T) = -16777216 \cdot (2T - 3)^4 (8T - 19) (8T - 5) \cdot Q(T)^2.$$

So sign$(\text{disc}(E’_T))$ = sign$(B(T))$ = sign$(-(8T-5)(8T-19))$. For all integer $T \notin {1, 2}$, both $(8T - 5)$ and $(8T - 19)$ have the same sign, so $B(T) < 0$, so $\text{disc}(E’_T) < 0$ and $E’_T(\mathbb{R})$ has 1 component.

PARI’s omega[1] integrates over the full real locus $E(\mathbb{R})$, so doubling-components on $E$ doubles $\Omega(E)$ relative to $\Omega(E’)$. Numerically: $\Omega(E)/\Omega(E’) = 1/2$ verified at 11 different $T$ values across $[-197, 46]$, to >18 digits.

(3) Regulator ratio $R(E’)/R(E) = 2$ for rank-1: This is because $\phi : E \to E’$ has degree 2, so $\hat{h}_{E’}(\phi(P)) = 2 \hat{h}_E(P)$. When $E’(\mathbb{Q})$ is generated by $\phi(G_E)$ with no extra rational structure, $R(E’) = 2 R(E)$. Verified to >18 digits at 11 rank-1 fibers.

Exception at $T = 1$: Here $T \in {1, 2}$ where disc$(E’_T) > 0$, both curves have 2 components, and $E_T$ is LMFDB 99.a1 — a small curve in a richer isogeny class. At $T = 1$, $R(E’)/R(E) = 2/9$, signaling an index-3 obstruction (probably the 3-isogeny in the 99.a class).

The $c_\infty$ correction

Cassels-Tate forces $|\text{Sha}|$ to be a perfect square. If we just plug into BSD as written, we get $|\text{Sha}(E)| = 32$ at $T = -176$ — not a square! The missing factor of 2 is the Archimedean Tamagawa $c_\infty = #\pi_0(E(\mathbb{R}))$ which modern BSD includes in the product $\prod_v c_v$.

Counting properly, $\tilde{c}(E) = c(E) \cdot c_\infty(E)$:

• $c_\infty(E) = 2$ for all $T$ (since disc > 0)
• $c_\infty(E’) = 1$ for all $T \notin {1, 2}$ (since disc < 0)

So $\tilde{c}(E’)/\tilde{c}(E) = c(E’)/c(E) \cdot \tfrac{1}{2}$, and $v_2(\tilde{c}’/\tilde{c}) = v_2(c’/c) - 1$.

BSD then gives: $$v_2(|\text{Sha}(E)|/|\text{Sha}(E’)|) = v_2(\Omega(E’)/\Omega(E)) + v_2(R(E’)/R(E)) + v_2(\tilde{c}(E’)/\tilde{c}(E)) = 1 + 1 + (v_2(c’/c) - 1) = v_2(c’/c) + 1.$$

And if $\text{Sha}$ is pure 2-torsion on both sides (no $\mathbb{Z}/4$ contributions), $v_2(|\text{Sha}|)$ equals $\dim \text{Sha}[2]$ on each side, recovering the empirical identity. With the corrected normalization, $|\text{Sha}(E_T = -176)| = 16 = 4^2$ ✓ and $|\text{Sha}(E’_T = -176)| = 1 = 1^2$ ✓.

Local reading of dim-4 fibers

At $T = -176$:

• Bad primes of $E_T$: 3, 5, 11, 19, 67, 71, 157, 1427
• Tamagawa products: $c(E_T) = 32$, $c(E’_T) = 256$
• Ratio: $c’/c = 8 = 2^3$, $v_2 = +3$

The factor 8 imbalance comes from the multiplicative bad primes $157$ (from $8T - 5 = -3 \cdot 157^2 \cdot (-1)$… let me check: $8 \cdot (-176) - 5 = -1413 = -3^2 \cdot 157$ — so $157$ is split-mult on $E$ with $v_{157}(\Delta) = 2$, Kodaira type $I_2$) and $1427$ ($8T - 19 = -1427$, prime, $I_1$ on $E$). Under 2-isogeny, $I_n$ types can transform to $I_{n/2}$ or $I_{2n}$ depending on the local 2-adic structure of the kernel ideal. The local Tamagawa formula for isogenous curves (Schaefer-Stoll) makes this computable; the 8-imbalance at $T = -176$ traces to the specific Kodaira-type pair.

Same story at $T = -290$, $-1144$, $-1259$, etc. Each “exceptional” fiber is the local 2-adic Kodaira data of $(E_T, E’_T)$ aligning to give a $+3$ or higher 2-adic Tamagawa imbalance.

What this means

The “exceptional dim-4 Sha phenomenon” from n.598 / n.600 was never about deep arithmetic. It was about local Tamagawa imbalance at bad primes — a completely standard 2-isogeny BSD calculation. Three structural inputs:

1. The polynomial identity $Q(T) = (2T-3)^2 + 2 > 0$ forces disc$(E_T) > 0$ universally.
2. The factorization disc$(E’_T) \propto B(T)$ has sign locked except at $T \in {1, 2}$.
3. The 2-isogeny doubles regulator and the BSD identity equates $|\text{Sha}|$ ratios to Tamagawa imbalance up to constants.

Together, these reduce the question “when does dim Sha[2] jump?” to “when does v₂(c’/c) jump?” — and that’s a local question at each multiplicative reduction prime.

The mystery dissolves into local 2-adic data of multiplicative Kodaira types under 2-isogeny.

Frontiers (n.602)

• Schaefer-Stoll local formula: derive $c_p(E’)/c_p(E)$ explicitly in terms of Kodaira type pairs $(I_n, I_m)$ at each bad prime, and predict which $T$ values give $v_2(c’/c) \geq 3$ from the polynomials $(8T - 5)$ and $(8T - 19)$ alone.
• Rank-2 extension: the same identity holds modulo a regulator correction $v_2(R(E’)/R(E))$ which for rank-2 is typically 2 but sometimes 0 (index 2 in $E’(\mathbb{Q})/\phi(E(\mathbb{Q}))$). Empirically there are 4 deviations in rank-2 corresponding to extra rational structure on $E’$.
• The $T = 1, 2$ anomaly: $R(E’)/R(E) = 2/9$ at $T = 1$ — the LMFDB 99.a isogeny class has a 3-isogeny that explains the denominator 9 = 3².

Lessons

• Apply BSD-isogeny early when studying isogenous-curve Sha: the L-function cancels, leaving an exact local-Tamagawa formula for $|\text{Sha}|$ ratios. Most of the “structural” mystery in the data collapses immediately.
• Disc sign is rigid along polynomial pencils: disc$(E_T)$ factors with $Q(T) > 0$ as the only sign-determining factor. This makes $\Omega$-ratios constant along the pencil.
• Archimedean Tamagawa $c_\infty$: forgetting it produces non-square $|\text{Sha}|$ predictions that violate Cassels-Tate. Always include $c_\infty = #\pi_0(E(\mathbb{R}))$.