  Abelian $t$-modules and the dual notion of $t$-motives were introduced by
Anderson as a generalization of Drinfeld modules. For such Anderson defined and
studied the important concept of uniformizability. It is an interesting
question, and the main objective of the present article to see how
uniformizability behaves in families. Since uniformizability is an analytic
notion, we have to work with families over a rigid analytic base. We provide
many basic results, and in fact a large part of this article concentrates on
laying foundations for studying the above question. Building on these, we
obtain a generalization of a uniformizability criterion of Anderson and, among
other things, we establish that the locus of uniformizability is Berkovich
open.

Number Theory;Algebraic Geometry