On the rank-3 singular Hecke order, the basis {1, T_3, T_11} is integral and has determinant valuation 1 in the ambient Q_3^3. Every Heegner operator is therefore a Z_3-linear combination of these three generators.
Reducing the coordinates modulo 3 recovers the syndrome-algebra formulas exactly:
T_3 = ET_7 = -I + NT_11 = NT_19 = -ET_43 = I + E - NT_67 = I + E - NT_163 = -I - E| operator | coords mod 3 in {1,T3,T11} |
coords mod 9 | coords mod 27 | first correction from mod-3 template mod 27 |
|---|---|---|---|---|
3 |
[0, 1, 0] |
[0, 1, 0] |
[0, 1, 0] |
[0, 0, 0] |
7 |
[2, 0, 1] |
[8, 6, 1] |
[8, 24, 10] |
[6, 24, 9] |
11 |
[0, 0, 1] |
[0, 0, 1] |
[0, 0, 1] |
[0, 0, 0] |
19 |
[0, 2, 0] |
[3, 2, 3] |
[12, 20, 3] |
[12, 18, 3] |
43 |
[1, 1, 2] |
[1, 7, 5] |
[10, 7, 5] |
[9, 6, 3] |
67 |
[1, 1, 2] |
[1, 4, 8] |
[19, 22, 26] |
[18, 21, 24] |
163 |
[2, 2, 0] |
[8, 5, 6] |
[8, 14, 24] |
[6, 12, 24] |
The mod-3 syndrome algebra is not an isolated shadow: it is the residue of an honest 3-adic order on the same generators. The first lift shows exactly where the node thickens:
T_43 and T_67 have the same mod-3 template, but distinct corrections modulo 9 and 27.T_11 stays the clean local axis: its coordinates are exactly [0,0,1] in every reduction.T_163 lifts the vacuum relation -I-E, with only 3-adic corrections in the nilpotent/local directions.So the syndrome algebra is the residue fiber of a genuine 3-adic Heegner algebra on {1, T_3, T_11}.