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7

Here is the kind of method I had in mind. We have the elliptic curve Kummer sequence $$0 \to E[n] \to E \to E \to 0,$$ Here I denote by $E[n]$ the $n$-torsion group scheme of $E$. Applying Galois cohomology we obtain $$0 \to E(\mathbb{Q})/nE(\mathbb{Q}) \to H^1(\mathbb{Q}, E[n]) \to H^1(\mathbb{Q}, E)[n] \to 0.$$ By the Mordell-Weil theorem, the group ...

0

The original question, as stated, has a negative answer. Namely, it is not true that the induced map $$\text{Hom}(A,A')\otimes\mathbb{Z}_p\to \text{Hom}_{F,V,\text{Fil}}(H^1_{\text{crys}}(\mathscr{A}'_k/W(k)),H^1_\text{crys}(\mathscr{A}_k/W(k)))\qquad (1)$$ is an isomorphism. For simplicity let $K=\mathbb{Q}_p$ (so $k=\mathbb{F}_p$). By the fully ...

2

Here's a bit of a long comment which I hope will help the OP. I also some of the questions that arose in the comments. Firstly, any smooth complex algebraic quasi-projective variety carrying an immersive period map is known to be "hyperbolic" in the sense that i) it is Brody hyperbolic, ii) all its subvarieties are of log-general type, and iii) the ...

4

Q2 should be yes for polarized VHS by a rigidity theorem of Schmid [theorem 7.24, Variations of Hodge structure, Inventiones 1973], which says roughly that a PVHS is determined by the Hodge structure at a fibre plus monodromy. Q1 should follow from this by Deligne's equivalence that you stated. Q3 is OK also. The point is that a group with trivial profinite ...

5

Here is a partial answer to questions 1/3. Let be given an abelian scheme $\mathscr A/V$. Let us assume that it carries a principal polarization; in principle, we may reduce to this case by an appropriate isogeny but I do not know whether there is a reference in the litterature. For every integer $n\geq 1$, the $n$-torsion subscheme $\mathscr A_n$ is finite ...

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