Local State-Machine Form of the Singular Hecke Module

Cyclic basis

On the singular arithmetic module L_sing, choose the cyclic generator w from the projected standard basis. Then

[ w , T_3 w , T_11 w ]

is a Z_3-basis, so every Heegner operator acts by a concrete 3 x 3 matrix in this basis.

This is the regular representation of the singular Hecke order on its arithmetic module.

Matrices modulo 3

• T_3:

[0, 0, 0]
[1, 1, 0]
[0, 0, 0]

• T_7:

[2, 0, 0]
[0, 2, 0]
[1, 0, 2]

• T_11:

[0, 0, 0]
[0, 0, 0]
[1, 0, 0]

• T_19:

[0, 0, 0]
[2, 2, 0]
[0, 0, 0]

• T_43:

[1, 0, 0]
[1, 2, 0]
[2, 0, 1]

• T_67:

[1, 0, 0]
[1, 2, 0]
[2, 0, 1]

• T_163:

[2, 0, 0]
[2, 1, 0]
[0, 0, 2]

First thickening modulo 27

• T_3 modulo 27:

[0, 18, 0]
[1, 4, 12]
[0, 3, 18]

• T_7 modulo 27:

[8, 0, 9]
[24, 8, 18]
[10, 9, 8]

• T_11 modulo 27:

[0, 0, 9]
[0, 12, 0]
[1, 18, 0]

• T_19 modulo 27:

[12, 9, 0]
[20, 20, 24]
[3, 6, 21]

• T_43 modulo 27:

[10, 18, 18]
[7, 17, 3]
[5, 3, 1]

• T_67 modulo 27:

[19, 18, 18]
[22, 14, 21]
[26, 21, 10]

• T_163 modulo 27:

[8, 9, 0]
[14, 1, 6]
[24, 15, 17]

Read

Modulo 3, these matrices recover the syndrome algebra in a cyclic module basis. Modulo 9 and 27, the matrices split the 43/67 collision and thicken the node without destroying cyclicity. The residue module already shows a nontrivial flag:

• the line spanned by T_3 w is the split semisimple branch;
• the line spanned by T_11 w is the nilpotent socle;
• the plane spanned by T_3 w and T_11 w is an invariant two-step memory layer.

So the singular Hecke factor is not just a ring defect: it acts as an explicit finite-state transition system on its own arithmetic module.