The Index-3 Overorder Above the Singular Local Hecke Factor

Two previously computed descriptions of the local factor looked inconsistent:

• the shifted coordinate relation suggested a quadratic algebra

  A_k = (Z/3^k Z)[y]/(y^2 - 9u_k),

  which is free of rank 2 over Z/3^k Z and therefore has size 3^(2k);

• the branch/conductor model gave the actual local Hecke order R_k of size 3^(2k-1).

The resolution is simple: R_k is an index-3 suborder of the quadratic overorder A_k.

modulus	k	size(A_k)	size(R_k)	index [A_k : R_k]
27	3	729	243	3
81	4	6561	2187	3
243	5	59049	19683	3
729	6	531441	177147	3

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The quadratic y-presentation is still useful, but it is not the integral Hecke order itself. It is a monogenic overorder sitting one F_3-dimension above the true local lattice. The conductor data shows the same index 3 gap at every tested finite level.

So the correct hierarchy is:

true local Hecke order   R_k
        ⊂
index-3 quadratic overorder   A_k = (Z/3^k Z)[y]/(y^2 - 9u_k)
        ⊂
split normalization   (Z/3^k Z) × (Z/3^k Z).

This resolves the apparent contradiction between:

• the branch model, which captures the actual integral order;
• the quadratic relation, which captures a convenient coordinate algebra in which the two branches are separated by the local generator.

So the singular Hecke node has three nested local models:

1. the true index-3 glued order,
2. a quadratic overorder with coordinate y,
3. the split normalization where the two branches are fully separated.