Perfect forms over imaginary quadratic fields

K. Scheckelhoff, K. Thalagoda, D. Yasaki

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. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Special Issue (9 - 2021), pp. 33-46