Abstract
I like to argue that crystalline cohomology will play a role in characteristic p analogous to the role of Ilodge theory in characteristic zero. One aspect of this analogy is that the F-crystal structure on crystalline cohomology should reflect deep geometric properties of varieties. This should be especially true of varieties for which the “p-adic part” of their geometry is the most interesting, as seems, often to be true of supersingular varieties in the sense of Shioda [26]. For example, in [19 §6] I proved that supersingular abelian varieties of dimension at least two are determined up to isomorphism by the F-crystal structure (and trace map) on H 1 cris , just as abelian varieties over ℂ are determined by the Ilodge structure on H 1 DR .
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Ogus, A. (1983). A Crystalline Torelli Theorem for Supersingular K3 Surfaces. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol 36. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9286-7_14
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DOI: https://doi.org/10.1007/978-1-4757-9286-7_14
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