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2

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 ...

3

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 ...

4

In this case, you can deduce without the Weil conjectures that $x_{i} \mapsto \displaystyle \frac{q}{x_{i}}$ permutes the eigenvalues. Let $f$ denote the Frobenius isogeny on $A$ and let $f^{\vee}$ be its dual isogeny. Then, we have $$f^{\vee}\circ f = q$$ which shows that if $x_{i}$ is an eigenvalue of $f$, then $\displaystyle \frac{q}{x_{i}}$ will be an ...

9

Let $B$ be a smooth projective curve over an algebraically closed field of characteristic zero. Let $K$ be the function field of $B$. Let $S$ be a finite set of closed points of $B$. You might find the following reformulation of Faltings's theorem less confusing. Theorem 1. (Faltings, geometric Shafarevich conjecture) Let $g$ be an integer. Then the set ...

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