4 problems

Let $K$ be a global function field of characteristic different from $p$, with absolute Galois group $G_K$. Let $n$ be a positive integer and let $\mathfrak{R}$ be an effective divi…

Let $K$ be a global function field with normalized valuation $v$, let $K_v$ be its completion, and let $\pi\in K_v$ satisfy $v(\pi)>0$. Let $A_v$ be the completion of the valuation…

Let $R$ be an excellent henselian discrete valuation ring with residue field $k$, and let $B=\operatorname{Spec}R$. Let $\mathcal Y$ be a regular scheme, flat and proper over $B$,…

Let $K$ be a global function field over a finite field $k$ of positive characteristic, let $S$ be a finite set of places of $K$, and let $\Gamma\subset O_S^*$ be a subgroup. Fix a…