Products of prime ideals in ray class groups

We prove that every class in the narrow ray class group modulo an integral ideal $\mathfrak q$ of a fixed number field is represented by a product of three prime ideals of norm at most $ ( N\mathfrak q)^{\max(1,3\alpha,4\alpha_0)+\kappa} $ for any $\kappa>0$, where $\alpha$ is the exponent in short character sum bounds for general non-principal ray class characters and $\alpha_0$ comes from a bounded-order subconvexity input for Hecke $L$-functions.

Wu's subconvexity bound gives the admissible choice $\alpha=\alpha_0=103/256$, hence the explicit bound $(N\mathfrak q)^{103/64+\kappa}$.

This improves the previous $O_K((N\mathfrak q)^3)$-scale bound of Deshouillers, Gun, Ramaré, and Sivaraman.

We also prove that a positive proportion of ray classes are represented by products of two prime ideals.

The proof extends the multiplicative dense-model and transference framework of Matomäki--Teräväinen to narrow ray class groups.