Undecidability of infinite algebraic extensions of Fₚ(t)

Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of Fₚ(t). As an application, we show that for every odd rational prime p there exist infinitely many primes r such that the fields Fpᵃ(t^r-∞) have undecidable first-order theory in the language of rings without parameters. Our method uses character theory to construct families of non-isotrivial elliptic curves whose Mordell-Weil group is finitely generated and of positive rank in Zᵣ-towers.