Arithmetic of \(K3\) surfaces (Q1018277)

This paper reviews the recent developments on the arithmetic of \(K3\) surfaces with special emphasis on (a) Modularity and singular \(K3\) surfaces; (b) Picard number and Galois action on the Neron-Severi group; and (c) Rational points and potential density. Results and ideas are formulated and illustrated for \(K3\) surfaces defined over \({\mathbb Q}\) or \({\mathbb F}_p\), for the sake of simplicity. (a) The author reviews \textit{R. Livné}'s theorem [Isr. J. Math. 92, No. 1--3, 149--156 (1995; Zbl 0847.11035)] on the modularity of singular \(K3\) surfaces defined over \({\mathbb Q}\). A complex \(K3\) surface with Picard number (the rank of the Néron--Severi group) \(\rho(\bar{X})=20\) is called \textit{singular}. Let \(T(X)\) denote the transcendental lattice of \(X\). Livné's Theorem: Let \(X\) be a singular \(K3\) surface defined over \({\mathbb Q}\) with discriminant \(d\). Then \(X\) is modular. That is, there is a newform \(f\) of weight \(3\) with CM by \({\mathbb{Q}}(\sqrt{d})\) and Fourier coefficients \(a_p\) of \(f\) such that for almost all prime \(p\), \({\text{trace}}({\text{Frob}}^*_p:T(X))=a_p\). Mazur and van Straten asked the converse question: \textit{Is every such newform of weight \(3\) associated to a singular \(K3\) surface over \({\mathbb Q}\)?} The answer is provided by \textit{N. D. Elkies} and \textit{M. Schütt} [Modular forms and \(K3\) surfaces, \url{arXiv:0809.0830}] for the \(65\) known imaginary quadratic fields with class group exponent two. Elkies--Schütt Theorem: Every known newform of weight \(3\) with Fourier coefficients \(a_p\in{\mathbb Z}\) is associated to a singular \(K3\) surface over \({\mathbb Q}\). These results are illustrated for Fermat quartic and its one-parameter deformation. (b) The determination of the Picard number is discussed. This is done comparing the Picard number \(\rho(\bar X)\) of a \(K3\) surface \(X\) over \({\mathbb Q}\) and the Picard number \(\rho(\bar{X}_p)\) of the reduction \(X_p\) of \(X\) modulo good prime \(p\). The Picard number \(\rho(\bar X)\) increases upon reduction. Van Luijk's criterion [\textit{R. van Luijk}, Algebra Number Theory 1, No. 1, 1--17 (2007; Zbl 1123.14022)]: Let \(X\) be a \(K3\) surface over \({\mathbb Q}\). Suppose that the Picard number \(\rho(\bar{X})\geq r\) with \(r\) odd. Assume that there are two good primes \(p_1\neq p_2\) such that the Picard number \(\rho(\bar{X}_{p_i})=r+1\) for \(i=1,2\). Let \(D=\text{disc}(NS(\bar{X}_{p_1}))/\text{disc}(NS(\bar{X}_{p_2}))\). If \(D\) is not a square in \({\mathbb{Q}}^{\times}\), then \(\rho(\bar X)\geq r\). Van Luijk's criterion requires the discriminant of the Néron--Severi group at two good primes, up to square. The discriminant may be computed using an algorithm of Kloosterman [\textit{M. Kuwata}, Comment. Math. Univ. St. Pauli 49, No. 1, 91--100 (2000; Zbl 1018.14013)], which invokes the Artin--Tate (conjectural) formula. Several applications of van Luijk's methods as well as one application of the Artin--Tate formula are discussed. (c) Rational points and density questions are discussed. Van Luijk's Theorem: In the moduli space of \(K3\) surfaces with a polarization of degree \(4\), the \(K3\) surfaces over \({\mathbb Q}\) with \(\rho=1\) and infinitely many rational points are dense. Theorem (Harris--Tschinkel) [\textit{J. Harris} and \textit{Yu. Tschinkel}, Duke Math. J. 104, No. 3, 477--500 (2000; Zbl 0982.14013)]: Let \(S\) be a smooth quartic in \({\mathbb{P}^3}\) over some number field \(k\). Assume that \(S\) contains a line \(\ell\) over \(k\) which does not meet more than five lines on \(S\). Then the set \(S(k)\) of \(k\)-rational points on \(S\) is dense in \(S\). If \(X\) is a variety over a number field \(k\), we say that the rational points are potentially dense on \(X\) if there is some finite extension \(k^{\prime}\) of \(k\) such that \(X(k^{\prime})\) is dense. Theorem (Bogomolov--Tschinkel) [\textit{F. A. Bogomolov} and \textit{Yu. Tschinkel}, Asian J. Math. 4, No. 2, 351--368 (2000; Zbl 0983.14008)]: Let \(X\) be a \(K3\) surface over a number field. Assume that \(X\) has an elliptic fibration or infinite automorphism group (so that \(\rho\geq 2\)). Then the rational points are potentially dense on \(X\). For \(\rho=1\), the only known result is the following. Theorem (Hassett--Tschinkel) [\textit{B. Hassett} and \textit{Yu. Tschinkel}, Am. J. Math. 130, No. 5, 1263--1278 (2008; Zbl 1152.14036)]: Let \(C\) be a complex curve. Then there are non-trivial \(K3\) surfaces over \({\mathbb{C}}(C)\) with \(\rho=1\) and Zariski-dense rational points.