Introduction

Let $N = 163$. The space $S_2(\Gamma_0(163))$ of weight-$2$ cusp forms has dimension $13$. The Hecke algebra $$\mathbb{T}_{163} = \mathbf{Z}[T_1, T_2, T_3, \ldots] \subset \mathrm{End}(S_2(\Gamma_0(163)))$$ is a free $\mathbf{Z}$-module of rank $13$. Its discriminant is $$\mathrm{disc}(\mathbb{T}_{163}) = 2^{15} \cdot 3^2 \cdot 65657 \cdot 82536739.$$ The $3$-adic valuation $v_3(\mathrm{disc}) = 2$ signals that $\mathbb{T}_{163} \otimes \mathbf{Z}_3$ is not a maximal order. The purpose of this paper is to give a complete description of the singular locus.

Over $\mathbf{Q}_3$, the algebra $\mathbb{T}_{163} \otimes \mathbf{Q}_3$ splits as a product of fields corresponding to the Galois orbits of newforms. Among the $13$-dimensional space, there is a unique non-semisimple $3$-adic local factor of rank $3$, involving two newform branches whose Hecke eigenvalues are congruent modulo $3$. We identify these branches, give the exact algebra presentation, and develop the full local theory: ideals, modules, and representations.

Context: Heegner discriminants

The number $163$ is the largest Heegner discriminant: $\mathbf{Q}(\sqrt{-163})$ has class number one. The seven Heegner discriminants $d \in \{3, 7, 11, 19, 43, 67, 163\}$ define Hecke operators $T_d$ on $S_2(\Gamma_0(163))$. Their behaviour on the singular factor is a motivating example throughout, though our results are purely local and do not depend on the Heegner property.

Main results

Our main contributions are:

1. An explicit presentation of the singular order (Theorem [thm:presentation]).

2. The identification of the glued newform pair as $V_1$ (rational) and one $\mathbf{Q}_3$-factor of the degree-$5$ orbit, with the Atkin–Lehner eigenvalue $U_{163} = -1$ characterising the node (Theorem [thm:branches]).

3. A complete ideal classification with closed-form local zeta function (Theorem [thm:ideals]).

4. The residue representation theory: two simples, one unique non-split self-extension (Theorem [thm:residue]).

5. A classification of all torsion-free (Cohen–Macaulay) modules: three indecomposables (Theorem [thm:CM]).

6. The singularity category with $2$-periodic suspension, Grothendieck group $\mathbf{Z}/2\mathbf{Z}$, and explicit stable Ext algebra (Theorem [thm:singcat]).

The singular order

Setup and notation

Let $\mathbb{T}= \mathbb{T}_{163} \otimes \mathbf{Z}_3$. The operator $T_2$ acts on the $13$-dimensional space and its characteristic polynomial factors over $\mathbf{Q}_3$ into irreducible pieces corresponding to Galois orbits of newforms. We write $V_1$ for the rational eigenform (the unique elliptic curve of conductor $163$), $V_5$ for a degree-$5$ orbit, $V_7$ for a degree-$1$ eigenform with $T_2$-eigenvalue $-1$, and the remaining dimensions for other orbits.

The congruence-order model

The full singular order

Branch identification

Geometry of the node

The Jacobson radical

The tangent cone

The syndrome algebra

Ideal classification and local zeta function

Residue representation theory

Cohen–Macaulay module classification

Fundamental exact sequences

The two branch modules are linked by the node:

The singularity category

In the singularity category $D_{\mathrm{sg}}(\Rpair) = \underline{\CM}(\Rpair)$, the free module $\Rpair$ becomes zero. What remains is the minimal non-trivial structure.

The stable Ext algebra

The Heegner operators in the local presentation

Using the identification $I = e_7 + e_{\mathrm{pair}}$, $E = e_7$, $N = -n$, the Heegner operators act on $\Osing$ as the following elements of $\mathbf{Z}_3 \times \Rpair$:

Family context

A scan of levels $N \le 199$ shows that levels $43$ and $67$ (the other large Heegner class-number-one discriminants) have trivial mod-$3$ syndrome quotient: $\ker(T_2^2) = 0$ over $\mathbf{F}_3$. Level $163$ is the first class-number-one level with a genuine three-dimensional syndrome quotient. The singular node described in this paper is therefore specific to level $163$ within the class-number-one family, though similar nodal structures can occur at other levels and primes.

Summary

The singular $3$-adic Hecke factor at level $163$ is completely described by one relation: $\eta^2 = 3\eta$. From this single quadratic identity over $\mathbf{Z}_3$, we derived:

1. the congruence-order model $\{(a,b) \in \mathbf{Z}_3^2 : a \equiv b \pmod{3}\}$;

2. the nodal geometry with tangent cone $g(g-a) = 0$;

3. the Jacobson filtration of constant width $3$;

4. the syndrome algebra $\mathbf{F}_3 \times \mathbf{F}_3[\varepsilon]/(\varepsilon^2)$ as exact mod-$3$ shadow;

5. the $\mathbf{P}^1(\mathbf{F}_3)$-bouquet of ideals at each depth;

6. the local zeta function $(1 + x + 3x^2)/(1 - x^2)$;

7. the residue Ext table with $\Ext^1(\chi_{\mathrm{pair}}, \chi_{\mathrm{pair}}) \cong \mathbf{F}_3$ as the unique non-semisimple direction;

8. the three CM indecomposables $B_0$, $\Rpair$, $B_3$ constituting the full module category;

9. the singularity category $D_{\mathrm{sg}}(\Rpair)$ with two objects, $2$-periodic suspension, and $K_0 \cong \mathbf{Z}/2\mathbf{Z}$;

10. the explicit Heegner operator images confirming the $T_{43}/T_{67}$ collision and $T_{11}$ as local generator.

The object is a two-state semisimple skeleton plus one nilpotent shadow, giving a three-state residue core with infinite $3$-adic refinement of constant width $3$. In the singularity category, even this reduces further: one bit, two branches, period $2$.

Computational verification. All results were verified in SageMath at precision $3^6 = 729$ using the full integral Hecke algebra at level $163$. The explicit data are recorded in Appendix 12.

Computational data

We record the exact computational inputs underlying the main theorems.

A.1. Characteristic polynomial of $T_2$

Over $\mathbf{Z}$, the characteristic polynomial of $T_2$ on the integral modular-symbols space $\mathrm{Symb}^1(\Gamma_0(163),\mathbf{Z})$ factors as $$\chi_{T_2}(x) = x \cdot (x^5 + 5x^4 + 3x^3 - 15x^2 - 16x + 3) \cdot (x^7 - 3x^6 - 5x^5 + 19x^4 - 23x^2 + 4x + 6).$$ The three factors correspond to the Galois orbits $V_1$ (degree $1$), $V_5$ (degree $5$), and $V_7$ (degree $7$), with $1+5+7=13=\dim V$.

Modulo $3$: $$\chi_{T_2}(x) \equiv x^3(x^4+2x^3+2)(x^6+x^4+x^3+x+1) \pmod{3}.$$ The $x^2$-primary component is $Q = \ker(T_2^2)$, of dimension $3$.

A.2. Restricted Hecke matrices on $Q$

In the adapted basis of $Q = \ker(T_2^2) \subset V_{\mathbf{F}_3}$, the seven Heegner operators act as the following $3\times 3$ matrices over $\mathbf{F}_3$: $$T_2|_Q = \begin{pmatrix}0&0&0\\0&0&0\\0&2&0\end{pmatrix}, \quad T_3|_Q = \begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}, \quad T_7|_Q = \begin{pmatrix}2&0&0\\0&2&0\\0&2&2\end{pmatrix},$$ $$T_{11}|_Q = \begin{pmatrix}0&0&0\\0&0&0\\0&2&0\end{pmatrix}, \quad T_{19}|_Q = \begin{pmatrix}2&0&0\\0&0&0\\0&0&0\end{pmatrix},$$ $$T_{43}|_Q = T_{67}|_Q = \begin{pmatrix}2&0&0\\0&1&0\\0&1&1\end{pmatrix}, \quad T_{163}|_Q = \begin{pmatrix}1&0&0\\0&2&0\\0&0&2\end{pmatrix}.$$ One verifies directly that $T_2|_Q = T_{11}|_Q$, that $(T_3|_Q)^2 = T_3|_Q$, $(T_{11}|_Q)^2 = 0$, and $T_3|_Q \cdot T_{11}|_Q = 0$.

A.3. The $3$-adic lift

Expressing the Heegner operators in the basis $(1, T_3, T_{11})$ of $A_H$ and lifting from $\mathbf{F}_3$ to $\mathbf{Z}/27\mathbf{Z}$:

The collision $T_{43} \equiv T_{67} \pmod{3}$ is visible in the first column. The two operators separate modulo $9$: the $T_{11}$-coefficients are $5$ and $8$ respectively. This separation is the $3$-adic manifestation of the nilpotent direction $\eta$ in the pair factor $R = \mathbf{Z}_3[\eta]/(\eta^2-3\eta)$.

A.4. Trace form on $Q$

On the $3$-dimensional algebra $A_H = \mathrm{Span}_{\mathbf{F}_3}\{I, E, N\}$: $$\operatorname{Tr}(I^2|_Q) = 3 \equiv 0 \pmod{3}, \qquad \operatorname{Tr}(E^2|_Q) = 1, \qquad \operatorname{Tr}(IE|_Q) = 1.$$ In particular, $\operatorname{Tr}(N \cdot X|_Q) = 0$ for all $X \in A_H$, confirming that the nilpotent direction lies in the radical of the trace form (rank $2$).