K3 surfaces of any Artin-Mazur height over $\mathbb{F}_5$ and $\mathbb{F}_7$ via quasi-$F$-split singularities and GPU acceleration

TL;DR

This paper introduces a rapid algorithm and GPU implementation to compute the Artin-Mazur height of Calabi-Yau hypersurfaces, demonstrating the existence of K3 surfaces with any height over finite fields of characteristic 5 and 7.

Contribution

It presents a novel fast computational method for Artin-Mazur height and applies it to show the existence of K3 surfaces with all possible heights over specific finite fields.

Findings

Existence of K3 surfaces with any Artin-Mazur height over $\\mathbb{F}_5$

Existence of K3 surfaces with any Artin-Mazur height over $\\mathbb{F}_7$

Implementation of GPU-accelerated algorithm for height calculation

Abstract

We develop a fast algorithm to calculate the Artin-Mazur height (equivalently, the quasi-$F$-split height) of a Calabi-Yau hypersurface, building on the work in arXiv:2204.10076. We provide a implementation of our approach, and use it to show that there are quartic K3 surfaces of any Artin-Mazur height over $\mathbb{F}_{5}$ and $\mathbb{F}_7$.