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.



finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme
Let $k$ be a finite field. Let $X$ be a normal proper variety. Let $G$ be a finite flat commutative group scheme over $X$ of order a power of $p$.


Lemma 1. If $T$ is a $G$-torsor over $X$ and $T$ is trivial over the generic point of $X$, then $T$ is trivial.


Proof. Namely, let $X' \subset T$ be the scheme theoretic closure of the rational section we get from the triviality of $T$ over the generic point. Then $X' \to X$ is a birational finite morphism, hence an isomorphism as $X$ is normal. QED


Lemma 2. Let $Z$ be a proper scheme over $X$. Then $G(Z)$ is finite.


Proof. We may pull back $G$ to $Z$. Then we see it suffices to show that given $\pi : Y \to Z$ finite flat, there are only a finite number of
sections $\tau$ of $\pi$. This is clear because a section is given by a
$\mathcal{O}\_Z$-algebra map $\pi\_\*\mathcal{O}\_Y \to \mathcal{O}\_Z$ and we have finiteness of
$H^0(Z, \mathcal{H}om(\pi\_\*\mathcal{O}\_Y, \mathcal{O}\_Z))$
as $Z$ is proper over the finite field $k$.


Lemma 3. Let $Y \to X$ be an alteration of proper varieties with $X$
normal. Then $\ker(H^1(X, G) \to H^1(Y, G))$ is finite.


Proof. If $Y \to X$ is a blow up, then the kernel is trivial by Lemma 1. There exists a blowup $X' \to X$ such that the strict transform $Y'$ of $Y$ is flat over $X'$ (google "flattening by blowing up"). Of course we may assume $X'$ is normalize (if not then just normalize $X'$). Combining these two steps we may assume $Y \to X$ is finite flat.


Assume $Y \to X$ is finite flat. Say $T \to X$ is a $G$-torsor which becomes trivial over $Y$. Choose a section $\sigma : Y \to T \times\_X Y$.
Using that $T \times\_X (Y \times\_X Y)$ is a $G$-torsor over $Y \times\_X Y$
we can take the "difference" between $\sigma \circ \text{pr}\_0$ and $\sigma \circ \text{pr}\_1$ over $Y \times\_X Y$ to get an element
$$
\tau = \sigma \circ \text{pr}\_0 - \sigma \circ \text{pr}\_1 \in
G(Y \times\_X Y)
$$
I leave it to the reader to see that $\tau$ determines the
isomorphism class of the torsor $T$ by descent theory for the
fppf covering $\{Y \to X\}$. By Lemma 2 there are only a finite
number of $\tau$. QED


Proposition. $H^1(X, G)$ is finite.


Proof. By Lemma 3 we may replace $X$ by an alteration. Hence we may assume that over the function field of $X$ we have a filtration of $G$ by closed
subgroup schemes such that the successive quotients have order $p$.
(This step requires you to know about finite group schemes over fields;
you can read about this in the book by Mumford about abelian
varieties for example.)
Any closed subgroup scheme of the generic fibre of $G$
extends to a finite flat closed subgroup scheme over
a blowup of $X$ by the same flattening techniques as used
in the proof of Lemma 3.
Thus finally we may assume there is a filtration
$$
(0) \subset G\_1 \subset \ldots \subset G\_{n - 1} \subset G\_n
$$
where $G\_i/G\_{i - 1}$ is a finite flat group scheme of order $p$.
In this way (using the long exact cohomology sequence) we reduce
to the case where $G$ has order $p$.


Now we use the classification of such group schemes over the normal
variety $X$ (you can easily deduce what I say from the Oort-Tate
paper). There are two cases.


The first case is where $G$ is a
closed subgroup scheme of a line bundle $L$ over $X$. In this case
there is a sequence
$$
0 \to G \to L \to L^{(p)} \to 0
$$
for some additive map as indicated. Since the cohomology of $L$
and $L^{(p)}$ are finite, we win.


The second case is where $G$ is a Galois twist of $\mu\_p$ in the
generic point. Here we can (after replacing $X$ by an alteration
again if necessary) assume that $G|\_U = \mu\_{p, U}$ for some
Zariski op $U$. Then we can use that $H^1(X, G) \subset H^1(U, G)$
by Lemma 1 and use the finiteness of $H^1(U, \mu\_p)$. QED


