Sifting for small split primes of an imaginary quadratic field in a given ideal class

Abstract

Let $D>3$, $D\equiv3\;(4)$ be a prime, and let $\mathcal{C}$ be an ideal class in the field $\mathbb{Q}(\sqrt{-D})$. In this article, we give a new proof that $p(D,\mathcal{C})$, the smallest norm of a split prime $\mathfrak{p}\in\mathcal{C}$, satisfies $p(D,\mathcal{C})\ll D^L$ for some absolute constant $L$. Our proof is sieve theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group $L$-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result.