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The singular 3-adic Hecke node at level 163

Paper III — Presentation, ideal theory, and module classification

Richard Hoekstra · March 2026 · PDF

η² = 3η

One relation. Six characters. Everything follows.

The singular order

The Hecke algebra T₁₆₃ at level 163 is a free Z-module of rank 13 with discriminant 2¹⁵ · 3² · 65657 · 82536739. The 3-adic valuation v₃(disc) = 2 signals a non-maximal order. This paper gives the complete local description of the singular locus.

Theorem (Presentation) The unique non-semisimple 3-adic local factor has rank 3 over Z₃:

O_sing ≅ Z₃ \times Z₃[η]/(η² - 3η)

The first factor is the free branch V₇. The pair factor R = Z₃[η]/(η² − 3η) carries the glued branches V₁ and V₅.

The pair factor admits the congruence description:

Congruence order: R ≅ {(a,b) ∈ Z₃² : a ≡ b mod 3}. The normalisation is Z₃ × Z₃. The conductor is f = 3Z₃ × 3Z₃. The normalisation defect has length 1.

The branches

The Atkin–Lehner eigenvalue U₁₆₃ separates the branches:

V₁ and V₅ (the node): U₁₆₃ = −1. The rational newform (elliptic curve 163a1) and one Q₃-factor of the degree-5 orbit. Congruent modulo 3, separated modulo 9 by the operator T₁₁.

V₇ (the free branch): U₁₆₃ = +1. The degree-7 orbit. Smooth. No singularity.

Geometry of the node

The spectrum Spec(R) is a split nodal curve over Z₃: two smooth branches meeting transversally at the closed point. The tangent cone is g(g−a) = 0 — two lines crossing at the origin. The Jacobson radical has constant width: dim(J^r/J^{r+1}) = 3 for all r ≥ 1. Modulo 3:

Residue algebra: O_sing / 3·O_sing ≅ F₃ × F₃[ε]/(ε²) — the syndrome algebra from Paper II.

Ideal classification

Between consecutive powers of the maximal ideal, there are exactly four intermediate ideals, indexed by P¹(F₃). The total number of ideals of R/3^k R is 5k − 3. The local zeta function is:

ζ_R(s) = (1 + x + 3x²) / (1 − x²),   x = 3^{−s}

Cohen–Macaulay modules

Theorem (CM classification) The torsion-free indecomposable R-modules are exactly:

B₀ = R/(η), R, B₃ = R/(η-3)

Every torsion-free module splits as B₀^u ⊕ R^c ⊕ B₃^v. There is no fourth indecomposable.

B₀ is branch V₁ (η acts as 0). B₃ is branch V₅ (η acts as 3). R is the node itself — the unique indecomposable that sees both branches. The category has finite type because R is a Bass order.

The singularity category

The two branches are linked by fundamental exact sequences:

0 \to B₃ \to R \to B₀ \to 0 and 0 \to B₀ \to R \to B₃ \to 0

Theorem (Singularity category) In D_sg(R), the free module R vanishes. Two indecomposable objects remain:

Σ(B₀) = B₃, Σ(B₃) = B₀, Σ² = Id

K₀(D_sg(R)) ≅ Z/2Z

The entire singularity decategorifies to one bit. North or south. B₀ or B₃. The two branches oscillate with period 2. The stable Ext algebra has four normal forms: u^m·e₀, u^m·e₃, u^m·x, u^m·y — where x and y are the odd generators and u is the degree-2 periodicity class.

The Auslander order and centre recovery

The endomorphism order of the CM generator M = B₀ ⊕ R ⊕ B₃ is an 8-dimensional algebra with rad³ = 0. Its quiver has three vertices with arrows through R only — no direct morphisms between B₀ and B₃.

Theorem (Centre recovery)

Z(End_R(B₀ \oplus R \oplus B₃)) ≅ R

The node recovers itself from its own representation theory.

The Heegner operators

All seven Heegner operators are identified explicitly in the local presentation. T₁₁ generates the nilpotent direction. T₄₃ ≡ T₆₇ mod 3 (the collision) and they separate modulo 9. T₁₆₃ acts as −1 on both branches of the node, consistent with U₁₆₃ = −1.

Computational data

The appendix records: the characteristic polynomial of T₂ over Z and modulo 3; all seven restricted Hecke matrices on Q; the 3-adic lift table modulo 3, 9, and 27; and the trace form values confirming the rank-2 claim. Every result is verified in SageMath at precision 3⁶ = 729.

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