Setup

We assume the gate-classification framework of . For an elliptic newform $f=\sum a_n q^n$ and a prime $p$, the gate type of $T_p$ on the $3$-adic formal-group quotient is determined by $a_p$ via $$\begin{array}{c|c} a_p & \text{type} \\\hline a_p=0 & A\text{ (annihilator)} \\ v_3(a_p)=0,\ a_p\equiv 1\pmod 3 & I\text{ (identity)} \\ v_3(a_p)=0,\ a_p\equiv 2\pmod 3 & S\text{ (swap)} \\ v_3(a_p)\geq 1 & H_{v_3(a_p)}\text{ (shift)} \end{array}$$ This depends only on $a_p$, hence only on the isogeny class of the underlying elliptic curve.

The class-number-one Heegner primes are $\{2,3,7,11,19,43,67,163\}$. Among these, the levels $N$ at which $S_2(\Gamma_0(N))\neq 0$ are $N\in H'=\{11,19,43,67,163\}$. At each such level the rational isogeny class is unique; let $f^N$ denote the corresponding newform ($11a$, $19a$, $43a$, $67a$, $163a$).

[def:sig] The Heegner gate set at level $N$ is $G_N=\{3,7,11,19,43,67\}\setminus\{N\}$, and the gate signature of $f^N$ is the map $$\omega_N \;:\; G_N \;\longrightarrow\; \{A,S,I,H_1\}, \qquad p \;\longmapsto\; \text{gate type of }T_p\text{ on }f^N.$$

The five signatures

[prop:table] Computed in SageMath, the gate types on the seven Heegner primes $\{2,3,7,11,19,43,67\}$ for each newform are:

(Entries marked $-$ are the self-prime $p=N$.) The Heegner gate signature $\omega_N$ is the row of this table restricted to $\{3,7,11,19,43,67\}\setminus\{N\}$.

Direct computation in SageMath of $a_p(f^N)$ followed by the classification of Definition [def:sig]. The script is in the appendix. Gate types are invariant on isogeny classes since they depend only on $a_p$.

The three theorems

[thm:distinct] The five gate signatures $\omega_{11},\omega_{19},\omega_{43},\omega_{67},\omega_{163}$ are pairwise distinct.

Compare any two rows of the table on the intersection of their gate sets. For instance, $\omega_{11}$ and $\omega_{19}$ both contain entries at $p\in\{3,7,43,67\}$, where they read $(S,I,H_1,S)$ and $(I,S,S,S)$ respectively — different at $p=3$. All ten pairs are similarly checked.

[thm:drift] For each $N\in H'$, define the drift rate on $G_N$ as $$\mu(N) \;=\; \frac{\#\{p\in G_N:\omega_N(p)=H_1\}}{\#\{p\in G_N:\omega_N(p)\neq A\}}.$$ Then $$\mu(11)=\tfrac14,\qquad \mu(19)=\tfrac15,\qquad \mu(43)=\tfrac12,\qquad \mu(67)=0,\qquad \mu(163)=\tfrac25.$$ The five values are pairwise distinct rationals in $[0,\,1/2]$.

Read off Proposition [prop:table], restricted to $G_N$:

• $N=11$, $G_{11}=\{3,7,19,43,67\}$, types $(S,I,A,H_1,S)$: $1$ shift, $1$ annihilator, $4$ non-annihilators, $\mu=1/4$.

• $N=19$, $G_{19}=\{3,7,11,43,67\}$, types $(I,S,H_1,S,S)$: $1$ shift, $0$ annihilators, $5$ non-annihilators, $\mu=1/5$.

• $N=43$, $G_{43}=\{3,7,11,19,67\}$, types $(I,A,H_1,I,H_1)$: $2$ shifts, $1$ annihilator, $4$ non-annihilators, $\mu=1/2$.

• $N=67$, $G_{67}=\{3,7,11,19,43\}$, types $(I,I,S,I,I)$: $0$ shifts, $0$ annihilators, $5$ non-annihilators, $\mu=0$.

• $N=163$, $G_{163}=\{3,7,11,19,43,67\}$, types $(A,S,H_1,H_1,I,I)$: $2$ shifts, $1$ annihilator, $5$ non-annihilators, $\mu=2/5$.

The five rates $\{1/4,1/5,1/2,0,2/5\}$ are distinct.

The conductor $163$ is not extremal in this family. The highest drift rate is $\mu(43)=1/2$ and the lowest is $\mu(67)=0$; $N=163$ sits at $\mu=2/5$, in the middle. The companion paper  treats the level-$163$ case in detail; the present table situates it within the family.

[thm:special]  

1. $N=67$ is the unique level in $H'$ at which $\omega_N$ contains neither a shift nor an annihilator. Equivalently, the $3$-adic shell coordinate of $f^{67}$ is frozen under any composition of gates in $G_{67}$.

2. On the extended set $\{2,3,7,11,19,43,67\}\setminus\{N\}$ (including $T_2$), $N=163$ is the unique level at which two gates are annihilators, namely $T_2$ and $T_3$.

For (i): the row of $67a$ on $G_{67}=\{3,7,11,19,43\}$ is $(I,I,S,I,I)$, containing only permutation types. All other rows contain at least one shift or annihilator entry. For (ii): the row of $163a$ has $T_2=A$ and $T_3=A$; the only other rows with any $T_2=A$ entry are $19a$ (one annihilator only) and the rest have $T_2\in\{I,S\}$.

Where the special primes land

[rem:annihilator-locations] Where the annihilators land across the extended Heegner family $\{2,3,7,11,19,43,67\}$:

• $f^{11}$: annihilator at $p=19$.

• $f^{19}$: annihilator at $p=2$.

• $f^{43}$: annihilator at $p=7$.

• $f^{67}$: no annihilator at any prime in the set.

• $f^{163}$: annihilators at $p=2$ and $p=3$.

Every annihilator that appears in the family lands at a Heegner prime, $p\in\{2,3,7,19\}$. We do not know whether this is structural or small-data noise.

[rem:shifts] Where the shifts land:

• $f^{11}$: one shift at $p=43$.

• $f^{19}$: one shift at $p=11$.

• $f^{43}$: two shifts at $p=11$ and $p=67$.

• $f^{67}$: no shifts.

• $f^{163}$: two shifts at $p=11$ and $p=19$.

The prime $p=11$ is a shift for the four newforms in which it is not the self-prime. No other prime in the gate set is shared as a shift across multiple levels.

What is not proved here

• No conceptual reason is given for the partition of drift rates, for why $N=67$ is frozen, or for why $N=43$ rather than $N=163$ achieves the maximum drift in the family.

• Theorem [thm:distinct] is a finite check, not a structural theorem. No claim is made that gate signatures are injective beyond this finite set.

• Remark [rem:annihilator-locations] (annihilators land at Heegner primes) is unproven and may be a coincidence at $n=5$.

• Remark [rem:shifts] (the prime $11$ is shift-distinguished) is unproven and may be a coincidence at $n=5$.

• No statement is made about the higher-shell types $H_2,H_3,\ldots$ These do not occur in the present family at the chosen primes; whether they occur at higher Heegner primes (none exist over $\mathbb{Q}$ beyond $163$) is moot.

Reproducibility

from sage.all import EllipticCurve

def v3(n):
    if n == 0: return None
    n = abs(n); k = 0
    while n % 3 == 0: n //= 3; k += 1
    return k
def unit3(n):
    return 0 if n == 0 else (n // (3**v3(n))) % 3
def classify(ap):
    if ap == 0: return 'A'
    v = v3(ap)
    return ('S' if unit3(ap) == 2 else 'I') if v == 0 else f'H{v}'

for label in ['11a1', '19a1', '43a1', '67a1', '163a1']:
    E = EllipticCurve(label); N = E.conductor()
    row = [classify(int(E.ap(p))) if p != N else '-'
           for p in [2, 3, 7, 11, 19, 43, 67]]
    print(label, row)

9 R. Hoekstra, The Hecke automaton at conductor $163$: monotonicity, $\mathbb{Z}/2\mathbb{Z}$ symmetry, and confinement on the formal-group quotient.

R. Hoekstra, The drift rate of the Hecke automaton at conductor $163$.