Products of prime ideals in ray class groups

Abstract

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.