Equivariant derived equivalence and rational points on \(K3\) surfaces (Q6160472)

Let \(X\) and \(Y\) be smooth \(K3\) surfaces over a nonclosed field \(K\). Suppose that \(X\) and \(Y\) are derived equivalent over \(K\), that is, there is an equivalence of bounded derived categories of coherent sheaves \(\Phi: D^b(X)\to D^b(Y)\), as triangulated categories, defined over \(K\). A derived equivalence respects many arithmetic properties. The main concern of this article is to consider whether or not \(X(K)\neq \emptyset\) if and ony if \(Y(K)\neq\emptyset\). The paper presents some results on this question in a very special case -- isotrivial families of \(K3\) surfaces over the pounctured disc. Let \(G=G_N\) be a finite cyclic group of order \(N\). Fix projective \(K3\) surfaces \(X\) and \(Y\) over \(\mathbf{C}\) with \(G\)-actions and consider the isotrivial families \(\mathcal{X},\,\mathcal{Y}\,\to \Delta_1:=\mathrm{Spec}(\mathbf{C}((t)))\) with generic fibers \(\mathcal{X}_t,\,\mathcal{Y}_t\) over \(K=\mathbf{C}((t))\). Theorem. Suppose that \(\mathcal{X}_t\) and \(\mathcal{Y}_t\) admit a derived equivalence \(\Phi: D^b(\mathcal{X}_t))\to D^b(\mathcal{Y}_t))\) over \(K\). If \(\mathcal{X}_t(K)\neq\emptyset\), then \(\mathcal{Y}_t(K)\neq\emptyset\). Proof is based on the analogy between equivariant geometry and descent for nonclosed fields. Isotrivial families over fields of Lautent series are linked to equivariant geometry, and proof is completed through analysis of fixed points. In particular, proof does not hinge on classification.