3-adic Lift of the Syndrome Algebra at Level 163

Singular basis

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.

Mod-3 reduction

Reducing the coordinates modulo 3 recovers the syndrome-algebra formulas exactly:

First 3-adic lift

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]

Read

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:

So the syndrome algebra is the residue fiber of a genuine 3-adic Heegner algebra on {1, T_3, T_11}.