Point counting on $K3$ surfaces and an application concerning real and complex multiplication

TL;DR

This paper develops p-adic algorithms for point counting on K3 surfaces to find explicit examples with real or complex multiplication, aiding the study of their arithmetic properties.

Contribution

It introduces efficient p-adic algorithms for point counting on K3 surfaces, facilitating the search for surfaces with real or complex multiplication.

Findings

Algorithms successfully count points on K3 surfaces over finite fields.

Enabled identification of K3 surfaces with specific multiplication properties.

Improved computational methods for arithmetic investigations of K3 surfaces.

Abstract

We report on our project to find explicit examples of $K3$ surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are $p$-adic in nature.