Undecidability of infinite algebraic extensions of $\mathbb{F}_p(t)$

TL;DR

This paper develops a framework to analyze the decidability of infinite algebraic extensions of function fields over finite fields, showing many such fields have undecidable first-order theories.

Contribution

It introduces a new method using character theory and elliptic curves to establish undecidability results for large classes of infinite algebraic extensions.

Findings

Existence of infinitely many primes r with undecidable fields $_{p^a}(t^{r^{- abla}})$

Construction of non-isotrivial elliptic curves with finitely generated Mordell-Weil groups

Framework applicable to broad classes of infinite algebraic extensions

Abstract

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 $\mathbb{F}_p(t)$. As an application, we show that for every odd rational prime $p$ there exist infinitely many primes $r$ such that the fields $\mathbb{F}_{p^a}\left(t^{r^{-\infty}}\right)$ 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 $\mathbb{Z}_r$-towers.