The main aim of this talk is to construct a canonical F-isocrystal $H(A)_K$ for an abelian scheme A over a p-adic complete discrete valuation ring of perfect residue field K. This F-isocrystal $H(A)_K$ comes with a filtration and admits a natural map to the usual Hodge sequence of A. Even though $H(A)_K$ admits a map to the crystalline cohomology of A, the F-structure on $H(A)_K$ is fundamentally distinct from the one on the crystalline cohomology. When A is an elliptic curve, we further show that $H(A)$ itself is an F-crystal and that implies a strengthened version of Buiumâ€™s result on differential characters. The weak admissibility of $H(A)$ depends on a modular parameter over the points of the moduli of elliptic curves. Hence the Fontaine functor associates a new p-adic Galois representation to every such weakly admissible F-crystal $H(A)$. This is joint work with Jim Borger.

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Last updated: 18 Sep 2024