On the bounded-conductor finiteness conjecture in equal characteristic

We investigate the equal-characteristic case of the Moon–Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n\ge 1$, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations $\rho :G_K\to {\mathrm{G}\mathrm{L}}_n({\bar{\mathbf{F}}}_p)$ that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields $K$ of characteristic $p$ admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation ${\rho }_r:G_K\twoheadrightarrow {\mathrm{S}\mathrm{L}}_2({\mathbf{F}}_{p^r})$ for each integer $r\ge 4$. As a corollary, we construct an everywhere unramified Galois extension $L\mathrm{/}K$, regular over ${\mathbf{F}}_p$, with Galois group $$\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}K)\cong \prod _{r\ge 4}{\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbf{F}}_{p^r})\mathrm{.}$$ Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic $p$, whereas they cannot exist over global function fields of characteristic different from $p$.