15 problems

Let $F$ be a number field, let $E$ be a coefficient field, and let $\Gamma_F$ denote the absolute Galois group of $F$. Consider a semisimple representation … which is unramified ou…

Let $K$ be a finite extension of $\mathbf{Q}_p$, let $\mathbf{F}$ be a finite field of characteristic $p$, and let $\overline{\rho}:G_K\to\operatorname{GL}_n(\mathbf{F})$ be absolu…

Let $F$ be a number field and let $\rho:G_F\to\operatorname{GL}_n(\overline{\mathbf{Q}_p})$ be as in the preceding conjectures. Let $\mathscr{T}[\rho]$ be the automorphically defin…

Let $\rho$ and the associated sets $\mathscr{P}\operatorname{ar}(\rho)$ and $\mathscr{T}[\rho]$ be as above. For a parameter $\delta$, let $W(\delta)$ be the subgroup and…

Let $F$, $\rho$, $N(\rho)$, and $\mathscr{X}[\rho]$ be as above. Let $\mathscr{T}$ be the rigid space of continuous characters of the diagonal torus, and let…

Let $F$ be a number field, let $\rho:G_F\to\operatorname{GL}_n(\overline{\mathbf{Q}_p})$ be continuous, absolutely irreducible, almost everywhere unramified, and odd at every real…

With $Y_\pi$, $X(\bar\rho,z)$, and the point $z$ as above, let $\widehat{\mathcal{O}}_{Y_\pi,z}$ and $\widehat{\mathcal{O}}_{X(\bar\rho,z),z}$ denote the completed local rings at…

Fix a prime $q\neq p$ splitting in $E$, choose $q'$ as in the source, and let $G$ be a unitary group in $d$ variables attached to $E/\mathbb{Q}$ satisfying the stated conditions at…

Let an arithmetic family of $p$-adic representations arise on an eigenvariety, with base scheme parametrizing the family. A global triangulation is an analytic variation of the tri…

Let $L$ be a finite extension of ${\mathbb Q}_p$, let $\mathscr{S}_{\mathrm{irr}}$ parameterize the two-dimensional irreducible trianguline representations, and for…

Kisin's conjecture. The canonical inclusion is an equality:

Triangulation conjecture. For every $x\in Z_{\mathrm{ncr}}$, such an affinoid neighborhood $U$ and triangulation exist, with the induced triangulation at $x$ determined by its cano…

Assume the representability conjecture for triangulordinary filtrations in rank $2$ and rank $1$, so that the morphism $p_\sigma\colon\mathscr{E}^0(\sigma)\to\mathscr{E}^0$ exists,…

Let $\mathscr{E}^0$ be the semistable, distinct-eigenvalue locus of the eigencurve, let $\mathscr{V}^0$ be its rank-$2$ family of Galois representations, and set … For each…

Let $\mathscr{X}/E$ be a reduced, separated rigid analytic space, and let $\mathscr{D}$ be a locally free sheaf of $(\varphi,\Gamma_K)$-modules over…