Perfect Forms over Imaginary Quadratic Fields

TL;DR

This paper computes perfect forms over all imaginary quadratic fields with discriminant up to 5000, analyzes the polytopes involved, and establishes bounds on their types and quantities, providing new insights into their structure.

Contribution

It systematically computes perfect forms for a wide class of imaginary quadratic fields and introduces bounds and heuristics for their combinatorial types and counts.

Findings

Bound on the combinatorial types of arising polytopes

Lower bound on the number of perfect forms

Heuristic for improved lower bounds in large discriminant cases

Abstract

In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that can arise regardless of discriminant and give a volumetric argument for a lower bound on the number of perfect forms as well as a heuristic for a better lower bound for imaginary quadratic fields of sufficiently large absolute discriminant.