The $16$-rank of $\mathbb{Q}(\sqrt{-p})$

TL;DR

This paper proves an unconditional density result for the 16-rank of class groups of imaginary quadratic fields, extending previous results that depended on a conjecture, thereby advancing understanding of class group distributions.

Contribution

The paper establishes the density result for the 16-rank of class groups of imaginary quadratic fields without relying on conjectural assumptions.

Findings

Unconditional proof of the 16-rank density result

Extension of previous conditional results to unconditional ones

Advancement in understanding class group distributions

Abstract

Recently, a density result for the $16$-rank of $\text{Cl}(\mathbb{Q}(\sqrt{-p}))$ was established when $p$ varies among the prime numbers, assuming a short character sum conjecture. In this paper we prove the same density result unconditionally.