5 problems

Let $X/F$ be a geometrically connected variety with base point $x\in X(F)$. For each $\u0**eell$, let $\hat{\mathcal{U}}_{\u0**eell,X,x}$ be the completed universal enveloping alge…

Let $X/F$ be a geometrically connected variety with a base point $x\in X(F)$, and let \pi_1^\u0**eell(X,x) be its ell-adic unipotent fundamental group viewed as a pro-unipotent r…

Let $X/F$ be a variety, let $i$ be a cohomological degree, and let H^i_\u0**eell(X) and $H^i_{c,\u0**eell}(X)$ denote the corresponding Weil--Deligne representations, with the la…

Let $F$ be a local field, let $X/F$ be a variety, and let $i$ be a cohomological degree. For each prime $ell$ different from the residue characteristic, consider the representation…

Serre's independence-of-ell conjecture. The tautological representation of $\mathbf{G}_\ell^\circ$ is independent of $\ell$. This predicts an $\ell$-independent algebraic descripti…