10 problems

Purity conjecture. The Weil–Deligne representation

Let $K$ be a nonarchimedean local field with residue characteristic $p$, let $X/K$ be a proper smooth variety purely of dimension $n$, let $\sigma \in W_K$, and let…

Let $Y$ be a smooth partially proper rigid analytic variety over $K$, and suppose that its Hyodo–Kato cohomology groups are of finite rank over $C^{\operatorname{nr}}$. These group…

Let $X$ be a proper, smooth algebraic variety over a number field ${\mathrm{E}}\subset\mathbb C$, and let ${\mathbf{G}}$ be the Mumford–Tate group of the Hodge structure…

Exceptional-pole conjecture. The point $s_0$ is an exceptional pole of $L(\pi\times\sigma,s)$ if and only if it is a pole of

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…

Let $F$ be a local field of residue characteristic $p>0$, let $X$ be a variety over $F$, and let $\ell\neq p$. By the $\ell$-adic monodromy theorem, one can attach a Weil–Deligne r…

For each embedding $\tau:K\hookrightarrow L$, let $HT_{\tau}=\{i_{1,\tau}<\cdots<i_{n,\tau}\}$ be $n$ distinct integers, and let $WD$ be an $n$-dimensional Weil–Deligne representat…

Let $K$ be a finite extension of $\mathbb Q_p$, let $\tau$ be an inertial type, and let $\widetilde\tau$ be a Frobenius-semisimple Weil–Deligne representation of $W_K$. Inertial lo…

Let $Fsubset K$ be a quadratic extension of local fields, let $G_n(K)=GL(n,K)$, and let $W'_K$ and $W'_F$ be the Weil–Deligne groups. For a finite-dimensional representation $\rho$…