The curve and its Hecke node

Let $N = 163$ and let $$E\colon y^2 + y = x^3 - 2x + 1$$ be the unique elliptic curve of conductor $N$ over $\mathbb{Q}$. It has rank $1$, trivial torsion, $|\text{\textcyr{Sh}}| = 1$, $j$-invariant $-96^3/163$, and generator $P = (1, 0)$ with canonical height $\hat{h}(P) = 0.1899\ldots$, satisfying $L'(E,1)/(\Omega \cdot \hat{h}(P)) = 1$ (BSD, verified numerically).

Write $f = \sum a_n q^n$ for the weight-$2$ newform attached to $E$ by modularity. The characteristic polynomial of $T_2$ on $S_2(\Gamma_0(163))$ factors over $\mathbb{Q}$ as $x \cdot g_5(x) \cdot g_7(x)$ with $g_5, g_7$ irreducible of degrees $5$ and $7$. The root $x = 0$ corresponds to $f$ (since $a_2(E) = 0$).

Over $\mathbb{Q}_3$, the factor $g_5$ acquires one linear root $\alpha_5$ with $\alpha_5 \equiv 12 \pmod{27}$, and $g_7$ acquires one linear root $\alpha_7$ with $\alpha_7 \equiv 3 \pmod{27}$. Together with $\alpha_f = 0$, these three roots give eigenforms $f_{V_1} = f$, $f_{V_5}$, $f_{V_7}$ spanning the singular $3$-adic Hecke factor.

The singular Hecke order is the rank-$3$ subring $$\mathcal{O}_{\mathrm{sing}} = \mathbb{Z}_3 \cdot 1 \oplus \mathbb{Z}_3 \cdot T_3 \oplus \mathbb{Z}_3 \cdot T_{11} \;\subset\; \mathbb{Z}_3^3$$ where the embedding into $\mathbb{Z}_3^3 = \mathbb{Z}_3^{V_7} \times \mathbb{Z}_3^{V_1} \times \mathbb{Z}_3^{V_5}$ sends each operator to its eigenvalue triple. The pair factor is the rank-$2$ projection $$R = \pi_{V_1,V_5}(\mathcal{O}_{\mathrm{sing}}) = \{(\sigma_1, \sigma_2) \in \mathbb{Z}_3^2 : \sigma_1 \equiv \sigma_2 \pmod{3}\} \;\cong\; \mathbb{Z}_3[\eta]/(\eta^2 - 3\eta),$$ with conductor $\mathfrak{c}= 3R$ and normalisation $\tilde{R} = \mathbb{Z}_3 \times \mathbb{Z}_3$. The conductor exact sequence is $$0 \to R \to \mathbb{Z}_3 \times \mathbb{Z}_3 \to \mathbb{F}_3 \to 0.$$

The congruence underlying the singularity is $a_p(f_{V_1}) \equiv a_p(f_{V_5}) \pmod{3}$ for all primes $p$. The form $f_{V_1} = f$ is rational; the form $f_{V_5}$ is defined over the number field $\mathbb{Q}[x]/(g_5(x))$ of discriminant $65657$.

The lattice action

Let $\Lambda = (\mathbb{Z}/L\mathbb{Z})^d$ be a $d$-dimensional periodic lattice. Assign Hecke operators $T_{p_1}, \ldots, T_{p_{2d}}$ to the $2d$ lattice directions, choosing from $\{T_3, T_7, T_{11}, T_{19}, T_{43}, T_{67}\}$.

The single-defect energy of $s \in R/3^k R$ is $$E(s) = 2d \cdot \|s\|^2 + \sum_{p \in \mathrm{dirs}} \|T_p \cdot s\|^2,$$ where $\|(\sigma_1, \sigma_2)\|^2 = |\sigma_1|_M^2 + |\sigma_2|_M^2$ with $|\cdot|_M$ the centered residue modulo $M = 3^k$.

In the normalisation, $T_p$ acts as $(\sigma_1, \sigma_2) \mapsto (\lambda_1(p)\,\sigma_1,\, \lambda_2(p)\,\sigma_2)$ where $\lambda_1(p) = a_p(f_{V_1})$ and $\lambda_2(p) = a_p(f_{V_5})$ are the eigenvalues on the two branches.

$E(s)$ is independent of the assignment of operators to directions.

$E(s) = 2d\,\|s\|^2 + \sum_{p} \|T_p \cdot s\|^2$ sums over all six operators regardless of pairing.

The vacuum sector

The residue classes $R/3R$ decompose into three sectors: $\mathcal{B}_0 = \mathfrak{c}/3\mathfrak{c}$ (vacuum) and $\mathcal{B}_1, \mathcal{B}_2$ (matter). Since $\lambda_1(p) = a_p(E) \in \mathbb{Z}$ for all $p$, elements of the conductor $\mathfrak{c}= 3R$ have the form $s = (3a, 3b)$ with $a, b \in \mathbb{Z}_3$.

[thm:vac] For $s = (3, 0) \in \mathfrak{c}$ the defect energy is $$E_0 = 9\Bigl(2d + \sum_{p} a_p(E)^2\Bigr) = 9 \cdot 135 = 1215 = 5 \cdot 3^5.$$ This value is independent of $k$ for $k \ge 4$. More generally, $E(3^n, 0) = 9^n \cdot 135$ for $1 \le n \le k - 3$.

$E(3,0) = 6 \cdot 9 + \sum_p |3\lambda_1(p)|^2$. Since $\lambda_1(p) = a_p(E)$ are ordinary integers, $|3a_p|_M = 3|a_p|$ once $M > 6\max|a_p|$, which holds for $k \ge 4$. The values $3a_p(E)$ for $p \in \{3,7,11,19,43,67\}$ are $\{0, 6, -18, -18, 21, -6\}$, giving $\sum (3a_p)^2 = 1161$ and $E_0 = 54 + 1161 = 1215$. The geometric tower follows by scaling.

The Hecke invariant is $$\sum_{p \in \{3,7,11,19,43,67\}} a_p(E)^2 = 0 + 4 + 36 + 36 + 49 + 4 = 129 = 3 \cdot 43,$$ factoring through the conductor prime $3$ and the Heegner prime $43$.

Confinement

[thm:conf] For $s \in R \setminus \mathfrak{c}$, write $s = (1 + 3a, 1 + 3b)$. Then the optimal energy satisfies $E_{\min}(k) = \Theta(3^{2k})$ as $k \to \infty$.

The leading term of $T_p \cdot s$ is $\lambda_2(p) \cdot 1$, which has $|\lambda_2(p)|_3 = O(3^k)$ since $\lambda_2(p)$ is an algebraic integer of degree $5$ over $\mathbb{Q}$ (not rational). The free parameters $(a, b)$ provide $2$ degrees of freedom to minimise $12$ squared terms. The system is overconstrained and the minimum is $\Theta(3^{2k})$.

[thm:z2] The spectra of $\mathcal{B}_1$ and $\mathcal{B}_2$ are identical at every $k$.

$(\sigma_1,\sigma_2) \mapsto (-\sigma_1,-\sigma_2)$ sends $\mathcal{B}_1 \leftrightarrow \mathcal{B}_2$ and preserves $E$. This realises the suspension $\Sigma^2 = \mathrm{Id}$ of $D_{\mathrm{sg}}(R)$.

The vacuum sector lifts to $\mathbb{Z}_3$ (the element $(3,0)$ is a fixed integer). The matter sector does not: the optimal element at precision $k$ changes at every $k$. The conductor $\mathfrak{c}= 3R$ is the sharp boundary: Hensel lifting converges on $\mathfrak{c}$ and fails on $R \setminus \mathfrak{c}$.

Two-defect interaction

[thm:two] Two vacuum excitations $s_a = s_b = (3, 0)$ at distance $r$ on $\Lambda$:

1. $V(r) = 0$ for $r \ge 2$. The defects are free.

2. $V(1) = -72 = -8 \cdot 3^2$ for nearest neighbours, independent of the direction assignment.

For $r \ge 2$, no link has both endpoints at defect sites, so $E_{\mathrm{pair}} = 2E_0$ and $V = 0$. For $r = 1$, the two shared directed links change from $\|s\|^2 + \|T_p s\|^2$ to $\|s - T_p s\|^2$ and vice versa. The difference sums to $V = -72$ over all six operators.

The vacuum algebra

In the normalisation, ring multiplication is componentwise: $(\sigma_1, \sigma_2) \cdot (\sigma_1', \sigma_2') = (\sigma_1\sigma_1', \sigma_2\sigma_2')$.

The boundary class is preserved: products of vacuum elements are vacuum elements. No product of two vacuum excitations produces matter.

$\sigma_1 \equiv 0$ and $\sigma_1' \equiv 0$ implies $\sigma_1\sigma_1' \equiv 0 \pmod{3}$.

$(3,0) \cdot (3,0) = (9, 0)$ with $E(9,0) = 9 \cdot E(3,0) = 10935$, exceeding the matter threshold. Two light vacuum excitations produce a heavy vacuum excitation.

$(3, 0) \cdot (0, 6) = (0, 0)$. A $\sigma_1$-excitation and a $\sigma_2$-excitation annihilate to the vacuum.

The boundary grammar

On the boundary $R/3R \cong \mathbb{F}_3$, each Hecke operator acts as multiplication by $a_p(E) \bmod 3$: $$\begin{array}{c|cccccc} p & 3 & 7 & 11 & 19 & 43 & 67 \\ \hline a_p \bmod 3 & 0 & 2 & 0 & 0 & 1 & 1 \end{array}$$ Three operators kill ($\times 0$: $T_3, T_{11}, T_{19}$), one swaps ($\times 2$: $T_7$), and two are the identity ($\times 1$: $T_{43}, T_{67}$).

On a plaquette with corners $b_{00}, b_{10}, b_{01}, b_{11} \in \mathbb{F}_3$, the boundary action cost $S_{\partial} = \sum_{\mathrm{links}} |b_i - (a_p \bmod 3) \cdot b_j|^2$ defines a weighted code on $\mathbb{F}_3^4$. The unique zero-cost pattern is the vacuum $(0,0,0,0)$. The cost function is invariant under the bulk precision $k$: the boundary grammar is a property of the conductor, not the bulk.

The Hecke automaton

The Heegner point $P = (1,0) \in E(\mathbb{Q})$ has formal logarithm $z(P) \in 3\mathbb{Z}_3$ with $z(P)/3 \equiv 1 \pmod{3}$ (matter sector). In the formal group $E_1(\mathbb{Q}_3) \cong \mathbb{Z}_3$, the six Hecke operators act by multiplication by $a_p(E)$.

[thm:automaton] The action of the Hecke eigenvalues $\{0, 2, {-6}, {-6}, 7, {-2}\}$ on $\mathbb{Z}_3$ by multiplication defines a one-counter automaton with states $(v, r) \in \mathbb{Z}_{\ge 1} \times \mathbb{F}_3^\times$ and transitions: $$\begin{array}{ll} \textup{ANNIHILATE}\; (a_3 = 0): &(v, r) \to \bot \\ \textup{SWAP}\; (a_7 = 2): &(v, r) \to (v,\, 2r) \\ \textup{SHIFT}\; (a_{11} = a_{19} = -6): &(v, r) \to (v{+}1,\, r) \\ \textup{ID}\; (a_{43} \equiv a_{67} \equiv 1): &(v, r) \to (v, r) \end{array}$$ The counter $v$ is monotone: it can only increase. The reachable set from the Heegner state $(1, 1)$ is exactly $3\mathbb{Z}_3 = \mathfrak{c}$, the conductor ideal. The matter sector $\mathbb{Z}_3^\times$ is unreachable.

The unit gates $\{2, 7, -2\}$ generate $(\mathbb{Z}/3^k\mathbb{Z})^\times$ at every $k$ (verified for $k \le 5$), giving full permutation within each shell. The shift gate $-6 = -2 \cdot 3$ increments $v$ by $1$ (the factor $-2$ is a unit, leaving $r$ unchanged modulo $3$). The annihilator sends everything to $\bot$. No gate decrements $v$, so the matter sector ($v = 0$) is unreachable.

The Heegner point $P \in E(\mathbb{Q})$ is a confined excitation of the Hecke lattice: it generates $E(\mathbb{Q})$ but lives in the matter sector of the defect spectrum, with energy diverging as $\Theta(3^{2k})$.

Numerical verification

The vacuum energy stabilises at $k = 4$. The matter energy grows as $E_m \approx 0.35 \cdot 3^{2k}$. The $\mathbb{Z}/2\mathbb{Z}$ symmetry is exact at every $k$.

The vacuum tower is exactly geometric with constant $135 = 6 + 3 \cdot 43$.

The probabilistic automaton

With uniform gate selection (probability $1/6$ each), the automaton becomes a Markov chain.

[thm:boltzmann] The steady-state depth distribution under uniform gate selection with continuous injection at $v = 1$ is geometric: $$\rho(v) \propto q^v, \qquad q = \frac{S}{A + S},$$ where $A$ is the number of annihilator gates and $S$ the number of shift gates. The permutation count $P$ does not enter. For $(A, S, P) = (1, 2, 3)$: $q = 2/3$.

Conditioned on survival (probability $(S{+}P)/(A{+}S{+}P)$ per step), each step shifts with probability $S/(S{+}P)$ and permutes otherwise. The depth increment is Bernoulli with parameter $S/(S{+}P)$. Combined with geometric killing (rate $A/(A{+}S{+}P)$), the steady-state depth is geometric with ratio $q = \text{(survival)} \times \text{(shift rate)} / (1 - \text{(survival)} \times \text{(1-shift rate)}) = S/(A+S)$.

The mean depth at death is $\langle v \rangle = 1 + S/A$. For $(1, 2, 3)$: $\langle v \rangle = 3$, corresponding to the $3$-shell ring $R/3^3 R = R/27R$.

The emergent temperature is $T = 1/\ln((A{+}S)/S) = 1/\ln(3/2)$ for the $(1,2,3)$ class. No temperature parameter is imposed: it follows from the gate statistics.

The light cone

The six Hecke gates classify the lattice directions: $$\begin{array}{ll} \text{Timelike (permutation, same shell):} & -x,\, \pm y \\ \text{Spacelike (shift, deeper shell):} & \pm z \\ \text{Null (annihilation):} & +x \end{array}$$ A vacuum soliton $(3, 0)$ propagates freely in the three timelike directions, irreversibly deepens in the two spacelike directions, and dies in the null direction. The signature $(3, 2, 1)$ is determined by the distribution of $v_3(a_p(E))$ over the six Heegner primes. The chirality ($+x$ null, $-x$ timelike) arises from the supersingularity $a_3(E) = 0$.

Lattice verification

A single defect $(3, 0)$ on a $24^3$ lattice at $\bmod\,729$ spreads with initial depth growth rate $$\frac{\Delta\langle v \rangle}{\Delta n}\bigg|_{n=0} = 0.40 = \frac{2}{5} = \frac{S}{S+P},$$ matching the automaton exactly. The depth saturates at $\langle v \rangle \approx 3.5$ due to $\bmod$-$M$ wrapping.

Universality

Among elliptic curves with $a_3(E) = 0$ and prime conductor $N < 1000$, five automaton classes arise:

Discussion

The lattice model is a sigma model with singular target. The target space $\mathrm{Spec}(R)$ is a nodal curve: two copies of $\mathrm{Spec}(\mathbb{Z}_3)$ meeting at a single point (the conductor). The field $s\colon \Lambda \to \mathrm{Spec}(R)$ assigns to each lattice site a section of the structure sheaf of the node.

The two branches are:

• $\sigma_1$: the rational newform $f = f_{V_1}$, attached to the elliptic curve $E\colon y^2 + y = x^3 - 2x + 1$;

• $\sigma_2$: the degree-$5$ eigenform $f_{V_5}$, defined over $\mathbb{Q}[x]/(x^5 + 5x^4 + 3x^3 - 15x^2 - 16x + 3)$.

The conductor $\mathfrak{c}= 3R$ is the singular point where the branches meet. The vacuum sector consists of sections supported at the node. The matter sectors consist of sections supported on a single branch. Confinement is the obstruction to extending a branch section to a global section of the nodal curve over the full $3$-adic ring.

The chain of identifications is: $$e^{\pi\sqrt{163}} \approx \mathbb{Z} \;\longrightarrow\; h(-163) = 1 \;\longrightarrow\; E/\mathbb{Q} \;\longrightarrow\; f \equiv g \pmod{3} \;\longrightarrow\; R \;\longrightarrow\; \text{confinement}.$$

The five numbers

The model is determined by five arithmetic constants: $$\begin{aligned} \textstyle\sum a_p(E)^2 &= 129 = 3 \cdot 43 & &\text{(vacuum stiffness, Heegner prime)} \\ E_0 &= 1215 = 5 \cdot 3^5 & &\text{(vacuum energy)} \\ V(1) &= -72 = -8 \cdot 3^2 & &\text{(nearest-neighbour attraction)} \\ E_m/M^2 &\to 0.35\ldots & &\text{(confinement constant)} \\ \Sigma^2 &= \mathrm{Id} & &\text{($\mathbb{Z}/2\mathbb{Z}$ of $D_{\mathrm{sg}}(R)$)} \end{aligned}$$

The discrete Poincaré disc

The automaton state $(v, r)$ has a natural polar structure: $v \ge 1$ is the radius (conductor depth), $r \in \mathbb{F}_3^\times$ the angle (branch choice). The Boltzmann factor $(2/3)^v$ defines a discrete metric $$ds^2 = \bigl(\tfrac{2}{3}\bigr)^v\,dr^2 + dv^2.$$ The angular cost decreases exponentially with depth: swapping branches is cheap in the bulk, expensive near the boundary. Geodesics prefer shallow orbits, but the shift gates push the soliton deeper at rate $2/5$ per step. The conductor is a repulsive force — the node at $v = 0$ pushes outward.

In the $3$-adic limit $k \to \infty$: the soliton drifts to $v = \infty$, angular motion costs nothing, and the soliton becomes a pure radial ray. Confinement is radial escape on the Poincaré disc.

The soliton

On the lattice, the vacuum excitation $(3, 0)$ is immortal: it never annihilates (the gate $T_3$ with $a_3 = 0$ contributes zero to the additive lattice sum, unlike the multiplicative automaton where it absorbs). It oscillates with period $3$ in amplitude and breathes with period $9 = 3^2$ in spatial extent, contracting to ${\sim}\,5$ sites every $9$ steps. The breathing period is the conductor squared.

The topological charge is $K_0(D_{\mathrm{sg}}(R)) = \mathbb{Z}/2\mathbb{Z}$: the $\sigma_1$-soliton $(3,0)$ carries charge $[B_0] = +1$, the $\sigma_2$-soliton $(0,6)$ carries $[B_3] = -1$, and ring multiplication gives annihilation: $(3,0) \cdot (0,6) = (0,0)$.

The path integral

The amplitude to reach lattice site $x$ after $n$ steps is the sum over all gate-sequences (paths) of length $n$ from the origin to $x$, weighted by the product of eigenvalues: $$G(x, n) = \sum_{\text{paths } \gamma: 0 \to x} \prod_{i=1}^{n} a_{p_i}(E).$$ Each eigenvalue $a_p$ contributes a direction-dependent mass: $$m(p) = \frac{\log |a_p(E)|}{\log 3}.$$ The six masses are:

One table. Three readings. The gate type ($v_3$) determines the automaton. The mass ($\log|a_p|$) determines the path integral. The eigenvalue itself ($a_p$) determines the lattice amplitude. All three are the same six integers $\{0, 2, -6, -6, 7, -2\}$, which are the Fourier coefficients of the elliptic curve $E\colon y^2 + y = x^3 - 2x + 1$ at the six Heegner primes.

Open questions

1. Is $\sum a_p(E)^2 = 3 \cdot 43$ a special value of the symmetric square $L$-function $L(\mathrm{sym}^2 E, s)$?

2. The Adèlic product over supersingular primes $p \in \{2, 3, 17\}$ gives a multi-tape automaton with constraint $v_3 \le v_2$ (the $3$-adic world nested inside the $2$-adic world). Is this constraint universal or specific to $E = \texttt{163a1}$?

3. The quantum automaton (Hamiltonian on the state space) has ground state at $v = V_{\max}/2$. The Heegner point multiple $4 \cdot 3^{v-1} \cdot P$ connects the quantum ground state to the formal group. Is this a Gross–Zagier relation?

4. The path integral gives direction-dependent mass $m(p) = \log|a_p|/\log 3$. Is the mass spectrum $\{0.63, 0.63, 1.63, 1.63, 1.77\}$ related to a spectral measure on the Hecke algebra?

5. The gate lens on neural networks. Classifying the weights of a trained network (e.g. GPT-2) by magnitude into kill/shift/permute bins reveals depth-dependent structure: shallow layers kill more ($A$ decreasing from $0.24$ to $0.12$), deep layers shift more ($S$ increasing from $0.29$ to $0.43$). Is this a universal property of trained deep networks?

6. The five automaton classes $(A, S, P)$ for curves with $a_3 = 0$ and prime conductor $N < 1000$: do they exhaust all possibilities, or are there further classes at larger $N$?