\(K3\) surfaces over finite fields with given \(L\)-function
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) \(\mathcal{X}\times_{\mathrm{Spec }\mathcal{O}_{\ell}}\mathrm{Spec }{\ell} \simeq X\times_{\mathrm{Spec }\ell}\mathrm{Spec }\ell\), (b) \(\mathcal{X}\) is regular, (c) the special fiber of \(\mathcal{X}\) is a reduced normal crossings divisor with smooth components, (d) \(\omega_{\mathcal X/\mathcal O_\ell}\simeq \mathcal{O}_{\mathcal{X}}\). The main result of this paper is to consider the converse question to Theorem 1. Given a function \(Z\) satisfying the conditions of Theorem 1, does there exist a \(K3\) surface whose zeta-function equals \(Z\)? Assuming semistable reduction, it is shown that the answer is yes if one allows a finite extension of the finite field. Theorem 2. Assume every \(K3\) surface \(X\) over a \(p\)-adic field satisfies Property \((\star)\). Let \[ L=\prod_{i=1}^{2d} (1-\gamma_i\,T)\in 1+T{\mathbb{Q}}[T] \] be a polynomial which satisfies the conditions of Theorem 1. Then there exists a positive integer \(n\) and a \(K3\) surface \(X\) over \({\mathbb{F}}_{q^n}\) such that \[ L_{\mathrm{trc}}(X/{{\mathbb{F}}_{q^n}},T)=\prod_{i=1}^{2d}(1-\gamma_i^n\,T). \] An important ingredient in the proof is: given \(L_{\mathrm{trc}}\), one constructs a \(K3\) surface over a finite field by first constructing a complex \(K3\) surface with complex multiplication by a suitably chosen CM field, then descends it to a number field, and finally reduces it to the residue field at a suitably chosen prime above \(p\). The existence of a complex \(K3\) surface with complex multiplication by a CM field \(E\) with \([E:{\mathbb{Q}}]\leq 20\) is proved.
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