On elliptic \(K3\) surfaces. (Q5954541)

The author classifies all possible configurations of singular fibers and the torsion parts of Mordell-Weil groups of complex elliptic \(K3\) surfaces, using the lattice-theoretic approach provided by the Torelli theorem for \(K3\). More precisely, let \(f: X\to \mathbb{P}^1\) be an elliptic \(K3\) surface and \(O\colon \mathbb{P}^1\to X\) a zero section. The torsion part of the Mordell-Weil group of \(f\) (denoted by \(G_f\)) is a finite abelian group. It is known that for a singular fibre \(f^{-1}(p)\), the cohomology classes of the irreducible components disjoint from the zero section \(O(p)\) span a negative definite root lattice in the Néron-Severi lattice of \(X\), which ist of type NEWLINE\( \tau(p):=A_l,\) \(D_m,\) or \(E_n\). Let \(R_f\subset \mathbb{P}^1\) be the locus of points \(p\) such that the fibers \(f^{-1}(p)\) are reducible. Then, the \(ADE\) type of \(f\) is defined as the formal sum NEWLINE\[NEWLINE\Sigma_f:=\sum_{p\in R_f} \tau(p).NEWLINE\]NEWLINE NEWLINEIn the paper under review, a complete list of such pairs \((\Sigma_f,G_f)\) is given. In the course of the computation, \textit{V.~V.~Nikulin}'s [Math. USSR, Izv. 14, 103--167 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111--177 (1979; Zbl 0408.10011)] theory of discriminant forms of even lattices over \(\mathbb{Z}\) is used. By applying a lemma of \textit{S. Kondo} [J. Math. Soc. Japan 44, No. 1, 75--98 (1992; Zbl 0763.14021)] and \textit{K. Nishiyama} [Jap. J. Math., New Ser. 22, No. 2, 293--347 (1996; Zbl 0889.14015)], it is possible to determine by a pure lattice-theoretic calculation which pairs actually occur for elliptic \(K3\). A computer-aided calculation gives the complete list of the 3279 configurations [available at \texttt{math.AG/0505140}], divided into sublists for the thirteen possible torsion groups.NEWLINENEWLINE2746 types correspond to a Mordell-Weil group without torsion, which are related (up to elementary transformations) to extremal elliptic \(K3\) surfaces.