Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface section of $S$. Let $\mathscr{A}/S$ be an Abelian scheme.

Are there good cases when the injection $\mathscr{A}(S)/\ell^n \hookrightarrow \mathscr{A}(C) /\ell^n$ is surjective?

If $\mathrm{rk} \mathscr{A}(C) = 0$, this holds true since the $\ell^n$-torsion subgroups of $\mathscr{A}(S)$ and $\mathscr{A}(C)$ are (always) isomorphic. $\mathrm{rk} \mathscr{A}(C) = 0$ is e.g. the case if $C \cong \mathbf{P}^1_k$ and $\mathscr{A}/C$ is constant.

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface section of $S$. Let $\mathscr{A}/S$ be an Abelian scheme.

Are there good cases when the injection $\mathscr{A}(S)/\ell^n \hookrightarrow \mathscr{A}(C) /\ell^n$ is surjective?

If $\mathrm{rk} \mathscr{A}(C) = 0$, this holds true since the $\ell^n$-torsion subgroups of $\mathscr{A}(S)$ and $\mathscr{A}(C)$ are (always) isomorphic. $\mathrm{rk} \mathscr{A}(C) = 0$ is e.g. the case if $C \cong \mathbf{P}^1_k$ and $\mathscr{A}/C$ is constant.

[Edit: Note that if $X$ is like $S$, but of arbitrary dimension, and $Y$ a smooth ample hypersurface section of dimension $\geq 2$, $\mathrm{rk} \mathscr{A}(X) = \mathrm{rk} \mathscr{A}(Y)$.]