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Elliptic fibrations of maximal rank on a supersingular K3 surface
T. Shioda Rikkyo University, Department of Mathematics, Tokyo, Japan
Аннотация:
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.
Ключевые слова:
$\mathrm{K3}$ surface, Mordell–Weil lattice, Artin invariant.
УДК:
512.7
MSC: 14J27,
14J28,
14H40 Поступило в редакцию: 26.06.2012
Язык публикации: английский
DOI:
10.4213/im8017