Rank 2 local systems and abelian varieties II

Let $X/\mathbb {F}_{q}$ be a smooth, geometrically connected, quasi-projective scheme. $\mathcal {E}$ semi-simple overconvergent $F$ -isocrystal on $X$ . Suppose that irreducible summands {E}_i$ of have rank 2, determinant $\bar {\mathbb {Q}}_p(-1)$ , and infinite monodromy at $\infty$ further for each closed point $x$ the characteristic polynomial is in $\mathbb {Q}[t]\subset \mathbb {Q}_p[t]$ Then there exists dense open subset $U\subset X$ such {E}|_U$ comes from family abelian varieties $U$ As an application, let $L_1$ lisse {Q}}_l$ sheaf has {Q}}_l(-1)$ all crystalline companions to exist (as predicted by Deligne's conjecture) if only scheme $\pi _U\colon A_U\rightarrow U$ $L_1|_U$ summand $R^{1}(\pi _U)_*\bar