Geodesic
Elliptic fibrations of maximal rank on a~supersingular K3 surface
Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 571-580

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We study a class of elliptic $\mathrm{K3}$ surfaces defined by an explicit Weierstrass equation to find elliptic fibrations of maximal rank on $\mathrm{K3}$ surface in positive characteristic. In particular, we show that the supersingular $\mathrm{K3}$ surface of Artin invariant 1 (unique by Ogus) admits at least one elliptic fibration with maximal rank 20 in every characteristic $p>7$, $p\ne 13$, and further that the number, say $N(p)$, of such elliptic fibrations (up to isomorphisms), is unbounded as $p\to\infty$; in fact, we prove that $\lim_{p\to\infty} N(p)/p^{2} \geqslant (1/12)^{2}$. Bibliography: 19 titles.
Keywords: $\mathrm{K3}$ surface, Mordell–Weil lattice
Mots-clés : Artin invariant. 
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     author = {T. Shioda},
     title = {Elliptic fibrations of maximal rank on a~supersingular {K3} surface},
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T. Shioda. Elliptic fibrations of maximal rank on a~supersingular K3 surface. Izvestiya. Mathematics , Tome 77 (2013) no. 3, pp. 571-580. http://geodesic.mathdoc.fr/item/IM2_2013_77_3_a7/