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

TL;DR

This paper presents a sieve theoretic proof that bounds the smallest split prime in a given ideal class of an imaginary quadratic field, avoiding complex zero-density estimates used in prior proofs.

Contribution

It introduces a new sieve-based method to bound the smallest split prime in an ideal class, bypassing the need for zero-density estimates and exceptional zero analysis.

Findings

Bound on smallest split prime: p(D,C) ^L

Sieve theoretic proof avoids zero-density estimates

Improved understanding of prime splitting in quadratic fields

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.