Equidistribution of quadratic roots in ray classes and Crocker's irreducible primes

Let O = Z[ω], ω = (−1 + i√3)/2. For a prime ideal 𝔭 not dividing 6, of residue degree one and norm q, let t𝔭 ∈ [0, q) be the root of 36T2 + 6T + 1 modulo q determined by 6t𝔭 ≡ ω (mod 𝔭). We prove that for any fixed nonzero integral ideal 𝔪 of O, any finite-order ray class character ψ modulo 𝔪 and any fixed integer h ≠ 0, the sum over prime ideals 𝔭 with N𝔭 ≤ Y, 𝔭 not dividing 6𝔪 and f(𝔭/Q) = 1 of ψ(𝔭)e(ht𝔭/N𝔭) is o(Y/log Y) as Y → ∞, so that the normalized roots t𝔭/N𝔭 are equidistributed in every fixed ray class. The proof reduces the sum to a shaped spectral (Type I) estimate for Poincaré series with nebentypus on Γ0(q0), proved from the automorphic Green function, the Kuznetsov formula and the Weil bound, a bilinear (Type II) estimate for ideal characters, and a prime sieve for complex sequences. For Crocker's irreducible primes, we record the stronger relative-density-zero implication of the Kurlberg–Luca–Shparlinski and Felix–Kurlberg fixed-point estimates. As an application of the ray-class theorem, the primes q ≡ 7 (mod 24) at which neither 3 nor 6 is a cube have relative density 1/18 among all primes, and those among them with neither an order-3 nor an order-6 representative divisible by its order have relative density 1/54. 2020 Mathematics Subject Classification: Primary 11N37; Secondary 11F72, 11R44, 11A07. The accompanying evidence archive contains the finite computations and Lean proof sources; the LaTeX source archive is also supplied.