The Singular 3-adic Hecke Factor at Level 163

First correction: T_2 is primitive over Q, but not integrally at 3

• T_2 has characteristic/minimal polynomial of degree 13
• discriminant of the full Hecke order: 1598161946303102976 = 2^15 * 3^2 * 65657 * 82536739
• discriminant of the monogenic order Z[T_2]: 163036280407189371350898287149056 = 2^15 * 3^8 * 65657 * 374083^2 * 82536739
• index [T : Z[T_2]] = 10100241 = 3^3 * 374083
• in particular, v_3([T : Z[T_2]]) = 3

So T_2 generates the Hecke algebra over Q, but the full integral 3-local fiber is not the naive monogenic order Z_3[T_2].

Rational 3-adic splitting

• factor degrees of the T_2 polynomial over Q_3: [1, 1, 1, 4, 6]
• so the rational 3-adic Hecke algebra splits as Q_3^3 × K_4 × K_6
• the singular part is the rank-3 factor coming from the three linear Q_3 roots reducing to 0 mod 3

Rank-3 singular order

• projecting onto the three singular Q_3 factors, the full 3-local order is spanned over Z_3 by {1, T_3, T_11}
• determinant valuation of this basis in the ambient Q_3^3: v_3 = 1
• every T_n for 1 <= n <= 13 has integral coordinates in this basis

Lifted idempotent and local rank-2 factor

• the mod-3 idempotent E = T_3|Q lifts uniquely modulo each 3^k
• splitting by e_k gives:
  • a rank-1 etale factor e_k R_k
  • a rank-2 local factor (1-e_k)R_k

In the local factor, with identity u0 = 1-e_k and generator n = (1-e_k)T_11, the relations are:

• modulo 27: local factor relation y^2 = 9
• modulo 81: local factor relation y^2 = 36
• modulo 243: local factor relation y^2 = 198
• modulo 729: local factor relation y^2 = 441

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

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

with u_k a unit. The first values are:

• modulo 27: u_k = 1
• modulo 81: u_k = 4
• modulo 243: u_k = 22
• modulo 729: u_k = 49

The high-precision 3-adic unit is:

u = 1 + 3 + 2*3^2 + 3^3 + 2*3^4 + 3^7 + 3^8 + 3^9 + 3^10 + 2*3^11 + 2*3^12 + 2*3^13 + 2*3^16 + 3^17 + 3^18 + 3^19 + 3^21 + 2*3^24 + 3^25 + 3^26 + 2*3^27 + 3^30 + 3^31 + 2*3^32 + 3^34 + 3^35 + 2*3^37 + 2*3^38 + 2*3^39 + 3^40 + 2*3^43 + 2*3^44 + 2*3^48 + 2*3^51 + 3^52 + 2*3^54 + 2*3^55 + 3^56 + 3^58 + 2*3^60 + 2*3^61 + 2*3^62 + 3^63 + 2*3^65 + 2*3^66 + 2*3^67 + 3^69 + 3^70 + 3^73 + 2*3^75 + 3^77 + 3^78 + 3^79 + 3^80 + 2*3^81 + 3^82 + 2*3^83 + 3^84 + 3^85 + 3^86 + 3^87 + 3^88 + 2*3^91 + 2*3^92 + 3^93 + 2*3^95 + 3^97 + 2*3^98 + 3^99 + 2*3^100 + 3^101 + 3^103 + 3^105 + 2*3^106 + 2*3^107 + 2*3^108 + 2*3^109 + 3^110 + 2*3^111 + 2*3^112 + 2*3^113 + 2*3^114 + 2*3^116 + 3^117 + 2*3^118 + 3^119 + 2*3^120 + 3^122 + 3^123 + 3^124 + 2*3^125 + 2*3^127 + 2*3^128 + 3^129 + 2*3^130 + 2*3^131 + 2*3^132 + 3^133 + 3^134 + 3^137 + 3^138 + 3^139 + 2*3^143 + 2*3^144 + 3^145 + 2*3^146 + 2*3^147 + 2*3^148 + 3^149 + 3^151 + 3^152 + 2*3^156 + 3^157 + 2*3^161 + 3^162 + 2*3^164 + 2*3^165 + 2*3^166 + 3^167 + 3^168 + 2*3^169 + 3^171 + 2*3^172 + 3^173 + 2*3^175 + 3^176 + 2*3^177 + 2*3^178 + 3^179 + 3^180 + 2*3^181 + 3^182 + 3^183 + 3^184 + 3^185 + 3^187 + 3^188 + 3^190 + 2*3^191 + 3^192 + 3^194 + 3^195 + 2*3^196 + 3^199 + 2*3^202 + 3^203 + 3^204 + 3^207 + 2*3^209 + 3^210 + 3^211 + 3^212 + 3^213 + 3^214 + 2*3^216 + O(3^218)

Local structure

• the rank-2 factor is genuinely local; its maximal ideal is generated by 3 and the local generator y
• over Z_3, the maximal ideal is not nilpotent, so the Loewy filtration is infinite
• in the finite quotients modulo 3^k, the ideal powers terminate; the first layers are recorded in the JSON output
• explicitly, for R_loc / 3^k R_loc one gets m^j = (3^j, 3^{j-1}y), hence
  • dim_{F_3}(m^j / m^{j+1}) = 2 for 1 <= j < k
  • dim_{F_3}(m^k / m^{k+1}) = 1 and m^{k+1} = 0
  • so the Loewy length of the quotient modulo 3^k is exactly k+1

Heegner operators inside the local factor

Two exact/stable features stand out:

• T_11 is the local generator by construction
• T_163 is numerically -u0 to the full working precision
• T_43 and T_67 coincide modulo 3, but split modulo 9; the canonical collision is a residue-field phenomenon, not a full 3-adic equality

Honest conclusion

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:

Z_3  ×  R_loc

where R_loc is a rank-2 local Z_3-order whose finite quotients are quadratic orders of the form (Z/3^k Z)[y]/(y^2 - 9u_k) with unit u_k.

So the non-semisimple mod-3 defect really comes from a quadratic 3-adic local ring sitting behind the syndrome algebra, not from the naive monogenic order Z_3[T_2].