Local ring refinement beyond mod 3

Beyond the mod-3 picture

Paper III describes the singular Hecke order modulo 3. This paper lifts the analysis to the full 3-adic integers, revealing structure invisible at the residue field level.

Discriminant computation

T₂ has characteristic/minimal polynomial of degree 13 (= dim S₂(Γ₀(163))). The discriminants are:

Rational 3-adic splitting

The T₂ polynomial factors over Q₃ with factor degrees [1, 1, 1, 4, 6]. The rational 3-adic Hecke algebra therefore splits as:

T ⊗ Q₃ = Q₃³ × K₄ × K₆

where K₄ and K₆ are unramified extensions of Q₃ of degrees 4 and 6 respectively. The singular part is the rank-3 factor coming from the three linear Q₃ roots that all reduce to 0 mod 3.

The rank-3 singular order

Projecting onto the three singular Q₃ factors, the full 3-local order is spanned over Z₃ by {1, T₃, T₁₁}. Every T_n for 1 ≤ n ≤ 13 has integral coordinates in this basis. The determinant valuation of this basis in the ambient Q₃³ is v₃ = 1.

Lifted idempotent and local factor

The mod-3 idempotent E = T₃|Q lifts uniquely modulo each 3^k. Splitting by e_k gives a rank-1 etale factor and a rank-2 local factor R_loc.

In the local factor, with identity u₀ = 1 − e_k and generator y = (1 − e_k)T₁₁, the local factor relations modulo successive powers of 3 are:

After the shift y = n − (b_k/2)u₀, the linear term vanishes exactly modulo each 3^k, and the local factor takes the monogenic form:

Over Z₃, the maximal ideal of R_loc is generated by 3 and y. It is not nilpotent (the Loewy filtration is infinite), but in the finite quotients modulo 3^k the ideal powers terminate with m^j = (3^j, 3^j−1y).

Key observations

T₁₁ as local generator. The generator of the local factor is T₁₁ (nilpotent escape operator), not T₃ (idempotent exclusion). This is consistent with the automaton picture: T₁₁ carries the depth information.

T₁₆₃ as negative identity. In the local factor, T₁₆₃ = −u₀ to full working precision. The Atkin-Lehner involution at the prime level acts as −1 on the singular part.

Canonical collision splits. T₄₃ ≡ T₆₇ (mod 3) — the canonical collision from Paper II — but they split modulo 9. The collision is a residue-field phenomenon, not a 3-adic one.

Summary

The singular 3-local Hecke factor is not just the mod-3 syndrome algebra. It is a semilocal rank-3 order that splits canonically into Z₃ × R_loc, where R_loc is a rank-2 local Z₃-order whose finite quotients are quadratic orders (Z/3^kZ)[y]/(y² − 9u_k) with unit u_k. The non-semisimple mod-3 defect comes from this quadratic 3-adic local ring, not from the naive monogenic order Z₃[T₂].