Canonical vector heights on \(K3\) surfaces -- a nonexistence result
The paper under review shows that there exist \(K3\) surfaces with no canonical vector height in the sense of \textit{A. Baragar} [Can. Math. Bull. 46, No. 4, 495--508 (2003; Zbl 1083.14517)]. This had been conjectured by \textit{A. Baragar} and \textit{R. Van Luijk} [Math. Comput. 76, No. 259, 1493--1498 (2007; Zbl 1109.14031)].NEWLINENEWLINEIn particular, if \(X\) is a variety defined over a number field \(K\), Baragar defined a vector height on \(X\) to be a function \(\mathbf{h}: X(\overline{K})\to\text{Pic}(X_{\overline{K}})\otimes\mathbb{R}\) satisfying:NEWLINENEWLINE(i) \(\mathbf{h}(\sigma(P))=\sigma_*\mathbf{h}(P)+\mathbf{O}(1)\) for any automorphism \(\phi\) of \(X\), and \(\mathbf{O}(1)\) is a bounded vector function, andNEWLINENEWLINE(ii) \(h_D(P)=\mathbf{h}(P)\cdot D + O(1)\) for any Weil divisor \(D\) and associated height function \(h_D\).NEWLINENEWLINEA vector height is said to be canonical if the error term \(\mathbf{O}(1)\) in (i) can be omitted.
scientific article; zbMATH DE number 6199274