\(K3\) surfaces over finite fields with given \(L\)-function
type: Item
instance of: scholarly article
title: \(K3\) surfaces over finite fields with given \(L\)-function
zbMATH Open document ID: 1344.14024
author: Lenny Taelman
published in: Algebra \& Number Theory
publication date: 2016-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.08547
review text: Let \(X\) be a \(K3\) surface over a finite field \(\mathbb{F}_q\) with \(q=p^a\) for some prime \(p\). Assume that \(X\) is not supersingular. Then its zeta-function is of the form \[ Z(X/\mathbb{F}_q,T)=[(1-T)L(X/\mathbb{F}_q,T)(1-q^2T)]^{-1} \] where \[ L(X/\mathbb{F}_q,T)=\mathrm{det}(1-\mathrm{Frob }T\mid H^2(X_{\mathbb F_q},\mathbb Q_\ell(1))\in\mathbb{Q}[T]. \] It is known that \[ L(X/\mathbb{F}_q,T)=L_{\mathrm{alg}}(X/\mathbb{F}_q,T)L_{\mathrm{trc}}(X/\mathbb{F}_q,T) \] and that \[ L_{\mathrm{alg}}(X/\mathbb{F}_q,T)=\prod_{\gamma_i\in\mu_\infty}(1-\gamma_i\,T)\quad L_{\mathrm{trc}}(X/\mathbb{F}_q,T)=\prod_{\gamma_i\not\in\mu_\infty} (1-\gamma_i\,T) \] where \(\mu_{\infty}\) is the group of complex roots of unity. The zeta function of \(X\) is known to have the following properties. Theorem 1: Let \(X\) be a \(K3\) surface over a finite field \(\mathbb{F}_q\) with \(q=p^a\) for some prime \(p\). Assume that \(X\) is not supersingular. Then (1) all complex roots of \(L_{\mathrm{trc}}(X/\mathbb{F}_q,T)\) have absolute value \(1\); (2) no roots of \(L_{\mathrm{trc}}(X/\mathbb{F}_q,T)\) is a root of unity; (3) \(L_{\mathrm{trc}}(X/\mathbb{F}_q,T)\in\mathbb{Z}_\ell[T]\) for all \(\ell\neq p\); (4) the Newton polygon of \(L_{\mathrm{trc}}(X/\mathbb{F}_q,T)\) has a slope sequence \(-a/h, -a, a/h\) with multiplicities \(h, 2d-2h, h\), respectively, where \(h\) and \(d\) are integers satisfying \(1\leq h\leq d\leq 10\); (5) \(L_{\mathrm{trc}}(X/\mathbb{F}_q,T)=Q^e\) for some \(e>0\) and some irreducible \(Q\in\mathbb{Q}[T]\), and \(Q\) has a unique irreducible factor in \({\mathbb{Q}}_p[T]\) with negative slope. Now define Property \((\star)\): A \(K3\) surface \(X\) over a finite extension \(k\) of \({\mathbb{Q}}_p\) is said to satisfy Property \((\star)\) if there exists a finite extension \(k\subset\ell\) and a proper flat algebraic space \(\mathcal{X}\to\mathrm{Spec }\mathcal{O}_{\ell}\) such that (a) \(\mathcal{X}\times_{\mathrm{Spec }\mathcal{O}_{\ell}}\mathrm{S…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 11G25
zbMATH DE Number: 6617181
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: zeta functions
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: finite fields
MaRDI profile type: Publication
OpenAlex ID: W3106399383
arXiv ID: 1507.08547
DOI: 10.2140/ANT.2016.10.1133
Recommended article: \(K3\) surfaces of finite height over finite fields
Recommended article: Finiteness results for \(K3\) surfaces over arbitrary fields
Recommended article: Unconditional construction of \(K3\) surfaces over finite fields with given \(L\)-function in large characteristic
Recommended article: K3 SURFACES OVER NUMBER FIELDS ANDl-ADIC REPRESENTATIONS
Recommended article: Q4668447
Recommended article: Ordinary K3 surfaces over a finite field
Recommended article: A family of \(K3\) surfaces and towers of algebraic curves over finite fields
Recommended article: On the Picard number of \(K3\) surfaces over number fields
Recommended article: K3 SURFACES OVER NUMBER FIELDS AND THE MUMFORD-TATE CONJECTURE
Recommended article: Q4432361
description: scientific article