Canonical Geometry of the Singular Hecke Node

The exact local pair factor is

R = Z_3[eta]/(eta^2 - 3 eta).

Equivalently,

R ≅ Z_3 ×_{F_3} Z_3
  = {(a,b) in Z_3 × Z_3 : a ≡ b mod 3}.

So the singular Hecke object is literally the split nodal curve over Z_3: two smooth Z_3-branches glued along the common residue point modulo 3.

Basic invariants

• Krull dimension: 1
• embedding dimension: 2
• complete intersection: yes
• Gorenstein: yes, because R is a hypersurface
• normalization: S = Z_3 × Z_3
• conductor: f = 3S = m
• overorders: exactly R ⊂ S

So there is only one genuine singularity here: a split A_1-type node.

Tangent cone

Let m = (3,eta). In the associated graded ring, write

a = [3] in m/m^2,
g = [eta] in m/m^2.

Since eta^2 = 3 eta, one gets

gr_m(R) ≅ F_3[a,g]/(g^2 - a g) = F_3[a,g]/(g(g-a)).

So the tangent cone is already the union of two crossing lines. The node is visible in first-order infinitesimal geometry.

Hilbert series

Because R/m has dimension 1 and each m^r/m^(r+1) has dimension 2 for r >= 1, the Hilbert series is

H_R(t) = 1 + 2t + 2t^2 + ... = (1+t)/(1-t).

For the full singular order

O_sing ≅ Z_3 × R,

the residue semisimple piece has dimension 2 and every deeper Jacobson layer has width 3, so

H_{O_sing}(t) = 2 + 3t + 3t^2 + ... = (2+t)/(1-t).

Local ideal zeta functions

For the pair factor, ideals of index 3^n occur with multiplicities

c_0 = 1,
c_{2r-1} = 1,
c_{2r} = 4   (r >= 1).

Hence the local ideal zeta function is

zeta_R(s) = 1 + sum_{r>=1} x^(2r-1) + 4 sum_{r>=1} x^(2r)
          = (1 + x + 3x^2)/(1 - x^2),

with x = 3^{-s}.

For the full singular order O_sing ≅ Z_3 × R, one more smooth branch contributes the factor (1-x)^{-1}:

zeta_{O_sing}(s) = (1 + x + 3x^2)/((1-x)(1-x^2)).

This is the cleanest current answer to the shape of the object: a smooth branch plus a split node, with completely explicit infinitesimal and ideal-theoretic geometry.