Rank 2 local systems and abelian varieties II

Let $X\mathrm{/}{\mathbb{F}}_q$ be a smooth, geometrically connected, quasiprojective variety. Let $\mathcal{E}$ be a semisimple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands ${\mathcal{E}}_i$ of $\mathcal{E}$ have rank 2, determinant ${\bar{\mathbb{Q}}}_p(-1)$, and infinite monodromy at $\mathrm{\infty }$. Suppose further that for each closed point $x$ of $X$, the characteristic polynomial of $\mathcal{E}$ at $x$ is in $$\mathbb{Q}$[t]\subset \mathbb Q_p[t]$\mathrm{.}Thenthereexistsanon-trivialopenset$U\subset X$ such that $\mathcal{E}{\mathrm{\mid }}_U$ comes from a family of abelian varieties on $U$. As an application, let $L_1$ be an irreducible lisse ${\bar{\mathbb{Q}}}_l$ sheaf on $X$ that has rank 2, determinant ${\bar{\mathbb{Q}}}_l(-1)$, and infinite monodromy at $\mathrm{\infty }$. Then all crystalline companions to $L_1$ exist (as predicted by Deligne’s crystalline companions conjecture) if and only if there exists a non-trivial open set $U\subset X$ and an abelian scheme ${\pi }_U\text{\hspace{0.17em}⁣}:A_U\to U$ such that $L_1{\mathrm{\mid }}_U$ is a summand of $R^1({\pi }_U{)}_{\ast }{\bar{\mathbb{Q}}}_l$.