7 problems

Let $\Delta=\{\alpha_1,\dots,\alpha_{n-1}\}$, and let $V$ be an object of $\operatorname{Rep}^{\operatorname{ord}}_h(G_{\mathbb{Q}_p,\Delta}\times\mathbb{Q}_p^\times)$. For each…

Let $p$ be an odd prime, let $V_{k,a_p}$ be a two-dimensional crystalline representation of $\operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and let $v_p(a_p)$ be a half…

Multiplicative comparison conjecture. The cohomology algebra

Let $P,P'\subset G$ be standard parabolic subgroups with standard Levi subgroups $L\subset P$ and $L'\subset P'$, and let $\pi$ and $\pi'$ be supersingular representations of…

Let $\mathfrak{m}$ be a maximal ideal of the Hecke algebra, with associated Galois representation $\bar{\rho}$. Let $\mathbf{F}[\mathfrak{m}]$ denote the corresponding Hecke-isotyp…

Let $B(\rho_x)^{\circ}$ and $\mathcal{B}_x^{\circ}$ be the lattices in the $p$-adic constructions, let $M^{\circ}$ be the specified lattice in…

Let $F$ be the relevant number field, let $K^p$ be a tame level with split ramification, and let $x\in\mathbb{X}(L)$ be a regular classical point such that $\rho_x$ is residually i…