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Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

Edit: I know it in the case $\mathscr{A} = A \times_k X$ is a constant Abelian scheme since then $H^1(X,\mathscr{A})[p]$ is finite and the Kummer sequence induces a short exact sequence $$0 \to \mathscr{A}(X)/p \to H^1_\mathrm{SYN}(X,\mathscr{A}[p]) \to H^1(X,\mathscr{A})[p] \to 0$$ and $\mathscr{A}(X)/p$ is finite by the Mordell-Weil theorem.

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

Edit: I know it in the case $\mathscr{A} = A \times_k X$ is a constant Abelian scheme since then $H^1(X,\mathscr{A})[p]$ is finite and the Kummer sequence induces a short exact sequence $$0 \to \mathscr{A}(X)/p \to H^1_\mathrm{SYN}(X,\mathscr{A}[p]) \to H^1(X,\mathscr{A})[p] \to 0$$ and $\mathscr{A}(X)/p$ is finite by the Mordell-Weil theorem.

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Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathcal{A}/X$$\mathscr{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathcal{A}[p]) = H^1_\mathrm{fppf}(X,\mathcal{A}[p])$$H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathcal{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathcal{A}[p]) = H^1_\mathrm{fppf}(X,\mathcal{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathscr{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathscr{A}[p]) = H^1_\mathrm{fppf}(X,\mathscr{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

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user19475

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathcal{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathcal{A}[p]) = H^1_\mathrm{fppf}(X,\mathcal{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma 8III.8.9.

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathcal{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathcal{A}[p]) = H^1_\mathrm{fppf}(X,\mathcal{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma 8.9.

Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathcal{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathcal{A}[p]) = H^1_\mathrm{fppf}(X,\mathcal{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.

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