2-adic point counting on \(K3\) surfaces
type: Item
instance of: scholarly article
title: 2-adic point counting on \(K3\) surfaces
author: Andreas-Stephan Elsenhans
author: Jörg Jahnel
published in: Research in Number Theory
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2202.10853
review text: The paper studies the problem of counting rational points of a class of \(K3\)-surfaces \(S\) defined over the finite field \(\mathbb F_p\) for an odd primt \(p\). By applying the Lefschetz trace formula, this is reduced to a statement on \(2\)-adic étale cohomology of said surfaces. Using the estimate \[ \lvert \# S(\mathbb F_p) - (p^2 +16p+1)\rvert \leq 6p, \] the authors conclude that it suffices to calculate \(\# S(\mathbb F_p)\) modulo a sufficiently large integer. Previous works due to Harvey and others give an algorithm to compute \(\# S(\mathbb F_p)\) modulo prime powers \(p^j\). The authors explain that while this algorithm is fairly efficient for computing this number modulo \(p^1\), it is already one order of magnitude slower for the modulus \(p^2\). They introduce a novel method to compute \(\# S(\mathbb F_p)\) modulo \(16.\). Combined with the above algorithm, this yields the number \(\# S(\mathbb F_p)\) modulo \(16p\), more than enough in light of the above estimate. Their algorithm relies on a very remarkable divisibility property for traces in the \(2\)-adic orthogonal group. The computational aspects of the paper seem to be done very diligently, implementing not only the paper's main algorithms but also methods from the background literature for comparison and validation. The data obtained for the primes up to \(10^8\) should be a treasure to search for new patterns. It is to be expected that the paper's code and data will appear shortly on its authors' websites, publicly available to the interested reader.
reviewed by: Felix Schremmer
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 7608398
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: point counting
zbMATH Keywords: 2-adic orthogonal group
zbMATH Keywords: 2-adic overdetermination
zbMATH Keywords: computational mathematics
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W4304144218
cites work: Supersingular $K3$ surfaces
cites work: The Magma algebra system. I: The user language
cites work: The Tate conjecture for \(K3\) surfaces over finite fields
cites work: Q4219187
cites work: On the distribution of the Picard ranks of the reductions of a \(K3\) surface
cites work: La conjecture de Weil. I
cites work: Cohomologie étale. Seminaire de géométrie algébrique du Bois-Marie SGA 4 1/2 par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier
cites work: Q3958572
cites work: Q4882944
cites work: Point counting on K3 surfaces and an application concerning real and complex multiplication
cites work: Computations with Algebraic Surfaces
cites work: Examples of \(K3\) surfaces with real multiplication
cites work: Q4195061
cites work: Q1233007
cites work: Computing zeta functions of arithmetic schemes
cites work: Computing Hasse–Witt matrices of hyperelliptic curves in average polynomial time, II
cites work: Q4542306
cites work: Q2778950
cites work: On a conjecture of Artin and Tate
cites work: Q4254917
cites work: Q3874320
cites work: Frobenius Maps of Abelian Varieties and Finding Roots of Unity in Finite Fields
cites work: Papers from the conference 21st Journées Arithmétiques held at the Università Lateranense, Rome, July 12--16, 2001
cites work: Cohomology and the Brauer Group of Double Covers
cites work: Some remarks on Brauer groups of \(K3\) surfaces
cites work: Modern computer algebra
cites work: The Finite Simple Groups
cites work: Discriminant of Certain K3 Surfaces
DOI: 10.1007/S40993-022-00378-X
Recommended article: Point counting on K3 surfaces and an application concerning real and complex multiplication
Recommended article: Counting rational points on K3 surfaces
Recommended article: COUNTING POINTS ON DWORK HYPERSURFACES AND -ADIC HYPERGEOMETRIC FUNCTIONS
Recommended article: Quantitative arithmetic of diagonal degree 2 \(K3\) surfaces
Recommended article: Counting rational points on Kummer surfaces
Recommended article: Counting rational curves on \(K3\) surfaces
Recommended article: K3 SURFACES OVER NUMBER FIELDS ANDl-ADIC REPRESENTATIONS
Recommended article: Counting elliptic curves in 𝐾3 surfaces
Recommended article: Counting points on varieties over finite fields related to a conjecture of Kontsevich
Recommended article: 𝑝-adic point counting on singular superelliptic curves
description: scientific article; zbMATH DE number 7608398