On the Finiteness of Geometric Representations for Varieties over Finite Fields

Let $p$ be a prime number, and let $k$ be a finite field of characteristic different from $p$. Let $X$ be a normal geometrically connected variety over $k$, let $\overline X$ be a compactification of $X$, and let $Z=\overline X\setminus X$. Let $D$ be an effective Cartier divisor on $\overline X$ whose support is contained in $Z$. Motivated by Hiranouchi's Hermite--Minkowski type theorem for varieties over finite fields, we formulate a finiteness conjecture for continuous semisimple geometric representations $$\pi_1(X,D)\longrightarrow \operatorname{GL}_n(F),$$ where $\pi_1(X,D)$ is Hiranouchi's fundamental group with ramification bounded by $D$, and $F$ is an algebraically closed field of characteristic $p$ endowed with the discrete topology. We prove this conjecture for odd $p$ in the following two cases: for curves with arbitrary ramification bound $D$, and for varieties of arbitrary dimension in the tame case, namely $D=0$. Furthermore, for arbitrary $p$, we prove the finiteness for those representations which admit a lift to characteristic zero.