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Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?

For example: If $K\ne \mathbb{C} $ and $X\rightarrow \operatorname{spec}(K)$ be a K3-surface. We do not have a Hodge isometry or a Period map (as far as I know). However we have etale cohomology and crystalline cohomology. If $K$ is a local field, then Falting's theorem on $p$-adic Hodge theory give us a Hodge-like decomposition of $$H^2_{et}(X_{K^{al}},\mathbb{Z}_p) \otimes \mathbb{C}_p \longrightarrow H^0(X,\Omega^2_X)(-2)\oplus H^1(X,\Omega^1_X)(-1)\oplus H^2(X,\Omega^0_X)(0)$$ as $G$-representations for $G$ the Galois group of $K^{al} /K$.

The theorem I am looking for is something kind of if we have a map $$\phi: H^2_{et}(X_{K^{al}},\mathbb{Z}_p) \longrightarrow H^2_{et}(X'_{K^{al}},\mathbb{Z}_p)$$ wich preserves the Hodge-like decomposition after tensoring with a Fontain's Ring (like $B_{dR}$), then there exists an isomorphism $f:X\to X'$ inducing $\phi$ on etale cohomology.

Any idea of a work/paper on this direction?

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    $\begingroup$ The Torelli theorem for supersingular K3 surfaces by Ogus is in terms of crystalline cohomology. Also, I believe the original Torelli theorem requires an integral Hodge isometry. If you tensor up with something is there still some "integral" structure and/or pairing to be preserved? (I don't know anything about Fontaine's rings). $\endgroup$
    – Matt
    Aug 3, 2013 at 22:54
  • $\begingroup$ Maybe this doesn't work for stupid reasons. Take $E, E'$ non-isomorphic isogenous of degree $n$ elliptic curves and $p=char(K)$. If $\ell\nmid np$, the isogeny induces an isomorphism as Galois modules $H^1_{et}(E_{k^{sep}}, \mathbb{Z}_\ell(1))$. If you make the Kummer K3's associated to the products $X=Km(E\times E)$ and $X'=Km(E'\times E')$ these might be "isogenous" in which on $\ell$-adic cohomology the "kernel" gets killed off (this is the part I don't remember). It should preserve all Galois rep properties and the integral structure yet not be isomorphic. $\endgroup$
    – Matt
    Aug 4, 2013 at 15:01
  • $\begingroup$ Thanks Matt. This paper of Ogus is good reference! I agree that the result may not be true just as I stayed. What I am looking for is a characterization of K3-surfaces in terms of its p-adic representations. If we Tensor $H^2(X,\mathbb Z)$ with $B_{dR}$ or $B_{HT}$ or $B_{st}$ etc.. we get different extra structures (Like Frobenius, monodromy, Filtrations etc…). It may not exist a result, but I was wondering if we can get something asking a morphism to preserve filtrations, or Frobenius or any of the extra structures we get after tensor with Fontain's rings. $\endgroup$
    – Rogelio Yoyontzin
    Aug 5, 2013 at 1:02
  • $\begingroup$ I meant $H^2_{et}(X,\mathbb Z_p)$ on my previews comment. $\endgroup$
    – Rogelio Yoyontzin
    Aug 5, 2013 at 22:46

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