Branch Model of the Singular 3-adic Hecke Factor

Main point

The rank-2 local factor is not the ring of integers in a quadratic field extension of Q_3. Its fraction algebra is already split: it lives inside Q_3 × Q_3.

What is singular is the integral order: two 3-adic branches are glued together modulo 3.

The branch coordinates below depend on a choice of square root of the quadratic relation. The resulting pair-order is canonical up to swapping the two branches.

Modulo 27

• the local factor is (Z/27Z)[y]/(y^2 - 9)
• square roots of 9 modulo 27: [3, 6, 12, 15, 21, 24]
• chosen branch root: 3

With this choice, the branch embedding is

a + b y  |->  (a + b r, a - b r)

and the image lands in the congruence order

{ (x_+, x_-) in (Z/3^k Z)^2 : x_+ == x_- mod 3 }.

Selected Heegner operator images: - T_11 -> [3, 24] - T_43 -> [25, 13] - T_67 -> [16, 13] - T_163 -> [26, 26]

Two stable features: - T_163 lands on the diagonal (-1,-1) modulo 3^k - T_43 and T_67 lie on different branches already modulo 9

Modulo 81

• the local factor is (Z/81Z)[y]/(y^2 - 36)
• square roots of 36 modulo 81: [6, 21, 33, 48, 60, 75]
• chosen branch root: 6

With this choice, the branch embedding is

a + b y  |->  (a + b r, a - b r)

and the image lands in the congruence order

{ (x_+, x_-) in (Z/3^k Z)^2 : x_+ == x_- mod 3 }.

Selected Heegner operator images: - T_11 -> [60, 48] - T_43 -> [31, 34] - T_67 -> [58, 25] - T_163 -> [80, 80]

Two stable features: - T_163 lands on the diagonal (-1,-1) modulo 3^k - T_43 and T_67 lie on different branches already modulo 9

Modulo 243

• the local factor is (Z/243Z)[y]/(y^2 - 198)
• square roots of 198 modulo 243: [21, 60, 102, 141, 183, 222]
• chosen branch root: 21

With this choice, the branch embedding is

a + b y  |->  (a + b r, a - b r)

and the image lands in the congruence order

{ (x_+, x_-) in (Z/3^k Z)^2 : x_+ == x_- mod 3 }.

Selected Heegner operator images: - T_11 -> [237, 195] - T_43 -> [7, 139] - T_67 -> [241, 4] - T_163 -> [242, 242]

Two stable features: - T_163 lands on the diagonal (-1,-1) modulo 3^k - T_43 and T_67 lie on different branches already modulo 9

Modulo 729

• the local factor is (Z/729Z)[y]/(y^2 - 441)
• square roots of 441 modulo 729: [21, 222, 264, 465, 507, 708]
• chosen branch root: 21

With this choice, the branch embedding is

a + b y  |->  (a + b r, a - b r)

and the image lands in the congruence order

{ (x_+, x_-) in (Z/3^k Z)^2 : x_+ == x_- mod 3 }.

Selected Heegner operator images: - T_11 -> [480, 438] - T_43 -> [250, 625] - T_67 -> [241, 490] - T_163 -> [728, 728]

Two stable features: - T_163 lands on the diagonal (-1,-1) modulo 3^k - T_43 and T_67 lie on different branches already modulo 9

Conceptual read

So the local factor is a glued pair of 3-adic branches:

R_loc  subset  Z_3 x Z_3

with the two coordinates constrained to agree modulo 3.

The residue-field collision T_43 ≡ T_67 (mod 3) is the first-order shadow of this gluing. Higher 3-adic precision resolves the two branches.

This is the clearest current picture of the defect:

• over Q_3: two distinct linear branches
• integrally: one singular order gluing them along residue mod 3
• modulo 3: the gluing collapses to the dual-number syndrome algebra