5 problems

Let $\rho$ be a motivic Galois representation with open image as above, let $m\geq 2$, and let $\mathcal{P}$ be a Cebatorev set, meaning that, up to finitely many primes, it is spe…

Let $h(z)$ be a degree-$n$ polynomial over a local field with residue-field size $q$, and let $\rho(n,\sigma;q)$ denote the density of polynomials whose associated finite étale alg…

Let $A$ be an abelian variety over a number field $F$, let $K/F$ be the extension appearing in the definition of the Chebotarev-Sato-Tate group, and let…

Let $L/k$ be a Galois extension of number fields with ${\mathrm {Gal}}(L/k) \cong G$. Let $\rho$ be a complex representation of $G$ of dimension $m$. Strong Artin conjecture. There…

Bhargava's Chebotarev splitting conjecture. For every prime $p$, positive integer $n$, and $\mu\in T_n^{\ast}$,