LetX/Fqbe a smooth geometrically connected variety. Inspired by work of Corlette-Simpson overC, we formulate a conjecture that absolutely irreducible rank 2 local systems withinfinite monodromy onX“come from families of abelian varieties”. WhenXis a projective variety,we prove that ap-adic variant of this conjecture reduces to the case of projective curves. If oneassumes a strong form of Deligne’s (p-adic)companions conjecturefrom Weil II, this implies that thel-adic version of our conjecture for projective varieties also reduces to the case of projective curves.Along the way we prove Lefschetz theorems for homomorphismsof abelian schemes and Barsotti-Tategroups. We also answer affirmitavely a question of Grothendieck on extending abelian schemes viatheirp-divisible groups.