For an elliptic newform f = Σ anqn and a prime p, the gate type of Tp on the 3-adic formal-group quotient is determined by ap alone:
| Condition on ap | Gate type | Effect on (r, k) |
|---|---|---|
| ap = 0 | A (annihilator) | (r, k) → ⊥ |
| v₃(ap) = 0 and ap ≡ 1 (mod 3) | I (identity) | (r, k) → (r, k) |
| v₃(ap) = 0 and ap ≡ 2 (mod 3) | S (swap) | (r, k) → (2r, k) |
| v₃(ap) ≥ 1 | Hv (shift) | (r, k) → (r, k + v) |
This depends only on ap, hence only on the isogeny class.
The class-number-one Heegner primes with non-zero cuspidal space are H' = {11, 19, 43, 67, 163}. At each level the rational isogeny class is unique (11a, 19a, 43a, 67a, 163a). Gate types on the extended Heegner primes {2, 3, 7, 11, 19, 43, 67}:
| fN | T₂ | T₃ | T₇ | T₁₁ | T₁₉ | T₄₃ | T₆₇ |
|---|---|---|---|---|---|---|---|
| 11a | I | S | I | -- | A | H₁ | S |
| 19a | A | I | S | H₁ | -- | S | S |
| 43a | I | I | A | H₁ | I | -- | H₁ |
| 67a | S | I | I | S | I | I | -- |
| 163a | A | A | S | H₁ | H₁ | I | I |
Entries marked -- are the self-prime p = N. The Heegner gate signature ωN is the row restricted to {3, 7, 11, 19, 43, 67} \ {N}.
The drift rate is μ(N) = #{p : ωN(p) = H₁} / #{p : ωN(p) ≠ A}, counting shifts over non-annihilators on the Heegner gate set GN.
| N | GN | Types on GN | Shifts | Non-ann. | μ(N) |
|---|---|---|---|---|---|
| 11 | {3,7,19,43,67} | (S, I, A, H₁, S) | 1 | 4 | 1/4 |
| 19 | {3,7,11,43,67} | (I, S, H₁, S, S) | 1 | 5 | 1/5 |
| 43 | {3,7,11,19,67} | (I, A, H₁, I, H₁) | 2 | 4 | 1/2 |
| 67 | {3,7,11,19,43} | (I, I, S, I, I) | 0 | 5 | 0 |
| 163 | {3,7,11,19,43,67} | (A, S, H₁, H₁, I, I) | 2 | 5 | 2/5 |
The five values {1/4, 1/5, 1/2, 0, 2/5} are pairwise distinct rationals in [0, 1/2].
Proof. The row of 67a on G₆₇ is (I, I, S, I, I). No entry is Hv for any v ≥ 1, so no gate changes the shell coordinate. All other rows contain at least one shift or annihilator entry.
Annihilators: f11 at p = 19; f19 at p = 2; f43 at p = 7; f67 nowhere; f163 at p = 2 and p = 3. Every annihilator in the family lands at a Heegner prime, p in {2, 3, 7, 19}. N = 163 is the unique level with two annihilators on the extended set.
Shifts: f11 at p = 43; f19 at p = 11; f43 at p = 11 and p = 67; f67 nowhere; f163 at p = 11 and p = 19. The prime p = 11 is a shift for the four newforms in which it is not the self-prime.