Let $U$ be an affine smooth surface over an algebraic closure of a finite field. Let $\mathscr{A}/U$ be an Abelian scheme and $\ell \neq \mathrm{char}(k)$ be prime.

Are there vanishing results for the étale cohomology with compact supports $H^2_c(U,\mathscr{A}[\ell^n]) = H^2(U,\mathscr{A}[\ell^n]^\vee(2))$ (the equality by Poincaré duality)? (My question is if there are $U$ such that this holds true, not if it is true for all $U$.)

Edit: $H^2(U,\mu_{\ell^n})$ is related to $\mathrm{Pic}(U)$ via the cycle class map.

Edit 2: What I want: Let $S/\bar{\mathrm{F}}_q$ be a smooth projective *surface*. Is there a (smooth?) projective ample hypersurface section $C \hookrightarrow S$ with affine complement $U \hookrightarrow S$ such that $H^2_c(U,\mathscr{A}[\ell^n]) = H^2(U,\mathscr{A}[\ell^n]^\vee(2)) = 0$?

My idea: Excise generators of $\mathrm{Pic}(S)/\ell^n$ and $\mathrm{Br}(S)[\ell^n]$ in order for $H^2(U,\mathscr{A}[\ell^n]^\vee(2))$ sitting in $$0 \to \mathrm{Pic}(U)/\ell^n \to H^2(U,\mu_{\ell^n}^{2g}) \to \mathrm{Br}(U)[\ell^n] \to 0$$ to vanish (the quotient of the Picard group $\mathrm{Pic}(S)/\ell^n$ is finitely generated; one could also assume $\mathrm{Br}(S)[\ell^n]$ to be finite). Can someone tell me if this works out correctly?