5 problems

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 $X$ 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 $K$ 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 $\rho$…