Abstract: Let k be an algebraically closed field of characteristic p≧ 0, and let X be an abelian variety over k. The goal of this paper is to answer the following questions, when dim(X) = 1 and p≠0, posed by R. Hartshorne: (1) Is E ( P ) indecomposable, when E is an indecomposable vector bundle on X ? (2) Is the Frobenius map F * : H 1 ( X, E ) → H 1 ( X, E (p) ) injective? We also partly answer the following question posed by D. Mumford: (3) Classify, or at least say anything about, vector bundles on X when dim ( X ) > 1.