Conductor of the Singular 3-adic Hecke Factor

Normalized branch model

After choosing a square root of the 3-adic unit in the local relation y^2 = 9u, the rank-2 local factor identifies with the congruence order

R_loc = { (a,b) in Z_3 × Z_3 : a ≡ b mod 3 }.

Its normalization is the split order

\tilde R = Z_3 × Z_3.

Conductor

The conductor ideal is

f = { (a,b) : a ≡ 0 mod 3 and b ≡ 0 mod 3 } = 3 Z_3 × 3 Z_3.

This is also the Jacobson radical / maximal ideal of R_loc.

modulo 27

• |S| = 729, |O| = 243, |f| = 81
• S/O has size 3 and F_3-dimension 1
• O/f has size 3 and F_3-dimension 1
• S/f has size 9 and F_3-dimension 2
• conductor definition verified directly: True
• T_67 - T_43 = [18, 0] in branch coordinates
• so T_67 - T_43 lies in the conductor: True

modulo 81

• |S| = 6561, |O| = 2187, |f| = 729
• S/O has size 3 and F_3-dimension 1
• O/f has size 3 and F_3-dimension 1
• S/f has size 9 and F_3-dimension 2
• conductor definition verified directly: True
• T_67 - T_43 = [27, 72] in branch coordinates
• so T_67 - T_43 lies in the conductor: True

modulo 243

• |S| = 59049, |O| = 19683, |f| = 6561
• S/O has size 3 and F_3-dimension 1
• O/f has size 3 and F_3-dimension 1
• S/f has size 9 and F_3-dimension 2
• conductor definition verified directly: True
• T_67 - T_43 = [234, 108] in branch coordinates
• so T_67 - T_43 lies in the conductor: True

modulo 729

• |S| = 531441, |O| = 177147, |f| = 59049
• S/O has size 3 and F_3-dimension 1
• O/f has size 3 and F_3-dimension 1
• S/f has size 9 and F_3-dimension 2
• conductor definition verified directly: True
• T_67 - T_43 = [720, 594] in branch coordinates
• so T_67 - T_43 lies in the conductor: True

Consequences

• The normalization defect has length 1: both S/O and O/f are one-dimensional over F_3.
• So the singularity itself is first-order, even though the 3-adic filtration by powers of the maximal ideal is infinite.
• The infinite tower comes from repeated multiplication by the conductor/maximal ideal f = m, not from a deeper normalization defect.
• The 43/67 collision is killed by the conductor: their difference lies in f, hence vanishes in the residue algebra but not in the normalized branches.