The Hecke automaton at conductor 163

The curve

Fix the elliptic curve E : y² + y = x³ − 2x + 1 of conductor 163 (Cremona label 163a1), with j-invariant −2¹⁸ · 3³ · 5³ · 23³ · 29³ / 163² and Mordell–Weil rank one, generator P = (0, 0). Write f = ∑ a_n qⁿ for its associated weight-2 newform.

For each prime p ≠ 163, the Hecke operator T_p acts on f by the scalar a_p ∈ Z. We study the six smallest such primes: p ∈ {3, 7, 11, 19, 43, 67}. These are the six gates of the automaton.

Gate classification

For an integer n ≠ 0, write v₃(n) for its 3-adic valuation and u₃(n) := (n / 3^v₃(n)) mod 3 ∈ F₃^× for its leading 3-adic unit. The gate type is determined by a_p as follows:

• If a_p = 0, then T_p is an annihilator.
• If a_p ≠ 0 and v₃(a_p) = 0, then T_p is a permutation gate with multiplier u₃(a_p) ∈ F₃^×. Multiplier 1 gives an identity; multiplier 2 gives a swap.
• If a_p ≠ 0 and v₃(a_p) ≥ 1, then T_p is a shift gate of depth v₃(a_p) and unit multiplier u₃(a_p).

Applied to conductor 163:

Proof: direct computation. 7 ≡ 1 (mod 3), so u₃(7) = 1 and T₄₃ is identity. −2 ≡ 1 (mod 3), so u₃(−2) = 1 and T₆₇ is identity. 2 ¬≡ 1 (mod 3), so T₇ is swap. −6 = −2 · 3 has v₃ = 1 and unit −2 ≡ 1 (mod 3), so T₁₁ and T₁₉ are shifts.

The state machine

The Hecke automaton A₁₆₃ has state space

Q = {VAC} ∪ { (r, k) : r ∈ F₃^×, k ∈ Z_≥0 }

with six input symbols {T_p : p ∈ {3, 7, 11, 19, 43, 67}} and transition function δ : Q × {T_p} → Q given by:

• δ(VAC, T_p) = VAC   (VAC is absorbing)
• δ((r, k), T₃) = VAC   (annihilator)
• δ((r, k), T_p) = (u₃(a_p) · r mod 3, k + v₃(a_p))   for p ≠ 3

Since u₃(a_p) ∈ F₃^× = {1, 2} for all p ≠ 3 and r ∈ F₃^×, the residue component never collapses to 0 under the non-annihilator gates. The only path into VAC from a unit state is through T₃.

Three theorems

Let Σ = {T₃, T₇, T₁₁, T₁₉, T₄₃, T₆₇} and Σ* = Σ \ {T₃}. For a word w = g₁ … g_n ∈ Σⁿ and a state q ∈ Q, write δ(q, w) for the iterated transition.

Proof. Every g ∈ Σ* has either v₃(a_p) = 0 (swap, identity) or v₃(a_p) = 1 (shifts T₁₁, T₁₉). The transition (r, k) ↦ (u₃(a_p) · r mod 3, k + v₃(a_p)) never produces VAC since u₃(a_p) ∈ {1, 2} and r ∈ {1, 2}. Iterating, the shell coordinate accumulates ∑ v₃(a_p(gi)) ≥ 0, with strict inequality iff some g_i ∈ {T₁₁, T₁₉}.

Proof. The residue acts by multiplication by u₃(a_p) ∈ F₃^×. From the gate table: u₃(a₇) = 2 while u₃(a_p) = 1 for p ∈ {11, 19, 43, 67}. Since F₃^× = {1, 2} has order 2, we have 2² = 4 ≡ 1 (mod 3).

Proof. Reachability: T₇ realises both residues from r = 1, and T₁₁^k reaches every shell k ≥ 0. Forward invariance of m: by Theorem 1, the shell coordinate never decreases, so once k ≥ 1 it remains ≥ 1. Absorption: both shift gates have v₃ = 1 > 0, so a single application sends k = 0 to k = 1.

The five numbers

What is not proved here

This paper makes only the claims stated above. In particular:

• We do not claim the formal-group orbit of the Mordell–Weil generator P ∈ E(Q) is monotone under multiplication-by-n. Numerical data show the shell pattern v₃(z(nP)) is not monotone in n beyond the first few rungs. The monotonicity proved here is a property of gate sequences acting on the state space, not of the integer orbit on the curve.
• We do not prove the automaton is universal among curves with similar Hecke data. The empirical universality classes observed across ten conductors are deferred.
• The connection to the singularity category D_sg(R) for R = Z₃[η]/(η² − 3η) is sketched only as motivation; the Z/2Z of Theorem 2 is established here purely as a property of residue arithmetic.

Reproducibility