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Let $$X/\mathbb {F}_{q}$$ X / F q be a smooth, geometrically connected variety. For X projective, we prove a Lefschetz-style theorem for abelian schemes of $$\text {GL}_2$$ GL 2 -type on X, modeled after a theorem of Simpson. Inspired by work of Corlette-Simpson over $$\mathbb {C}$$ C , we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on X come from families of abelian varieties. We have the following application of our main result. If one assumes a strong form of Deligne’s (p-adic) companions conjecture from Weil II, then our conjecture for projective varieties reduces to the conjecture for projective curves. We also answer affirmitavely a question of Grothendieck on extending abelian schemes via their p-divisible groups.

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

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The ethanolic extract of Hypnea muciformis (red algae) was tested for hepatoprotective activity against experimentally induced liver damage by Carbon tetrachloride (CCl 4