2-adic point counting on $K3$ surfaces

TL;DR

This paper introduces a novel 2-adic orthogonal group approach combined with p-adic methods to efficiently count points on K3 surfaces over finite fields for primes less than 10^8.

Contribution

It presents a new 2-adic orthogonal group technique integrated with p-adic methods for point counting on K3 surfaces over finite fields.

Findings

Successfully counted points on K3 surfaces for all primes p < 10^8

Developed a new approach combining 2-adic and p-adic methods

Enhanced efficiency in point counting over large finite fields

Abstract

This article reports on an approach to point counting on algebraic varieties over finite fields that is based on a detailed investigation of the $2$-adic orthogonal group. Combining the new approach with a $p$-adic method, we count the number of points on some $K3$ surfaces over the field $\bbF_{\!p}$, for all primes $p < 10^8$.