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 number field, let $n$ be a positive integer, let $\mathfrak{R}$ be a nonzero ideal of the ring of integers of $K$, and let $G_K$ be the absolute Galois group of $K$. F…

Let $G$ be the reductive group and $L$ its reflex field from the setup, let $K$ be the level, let $p$ be the coefficient characteristic, and let $\Sigma_{K,p}$ contain the places w…

Let $R(p)$ be the number of isomorphism classes of continuous representations … that are irreducible, odd, and unramified outside $p$. Let $U(p)$ be the upper bound for $R(p)$ supp…