15 problems

Pessimistic conjecture. The image of this map is closed in the Banach-space topology of $H^1(\mathcal{K}_\infty,V)$. This weaker assertion is stated as sufficient for the paper's p…

Let $F$ be the coefficient field, let $\mathcal{R}_F$ be its Robba ring, and let $\mathcal{D}$ be a $(\varphi,\Gamma)$-module over $\mathcal{R}_F$. For a representation $V$ of…

Let $F$ be the coefficient field, let $\mathcal{R}_F$ be its Robba ring, and let $\mathcal{D}$ be a $(\varphi,\Gamma)$-module over $\mathcal{R}_F$. The quotient…

Kedlaya's conjecture. For every $r>0$, the natural map

Kedlaya's conjecture. Locally analytic vectors should provide a nice $(\phi,\Gamma)$-module theory for any infinitely ramified $p$-adic Lie extension, and this theory should also b…

Emerton–Gee–Hellmann's locally analytic categorical Langlands conjecture. The functor $\mathfrak{LL}_p^{la}$ is fully faithful when restricted to a suitable category of “tempered”…

Nakayama's local epsilon conjecture. There exists an isomorphism

Nakayama's local epsilon conjecture. There exists a unique isomorphism

Equivalence conjecture. The functors $D_X$ and $D_X^{\operatorname{perf}}$ each define an equivalence of categories. This conjecture predicts that the comparison between the overco…

Let $R$ be a coefficient ring and let $D$ be an étale $(\varphi,\Gamma)$-module over $\mathcal{E}_R$, of rank $r_D$. The convolution pairing induces an alternating map … Using the…

Let $E$ be a finite extension of $\mathbf Q_p$, let $G_{\mathbf Q_p}=\mathrm{Gal}(\overline{\mathbf Q}_p/\mathbf Q_p)$, and let $\mathbf D$ denote Fontaine's equivalence from $p$-a…

Let $K$ be a finite extension of $\mathbf{Q}_p$, let $K_\infty/K$ be a Lubin–Tate extension, and let $(\varphi,\Gamma)$-modules be the modules attached to $p$-adic representations…

Let $K$ be the finite extension of the maximal unramified extension of a finite extension of $\mathbb{Q}_p$ considered in the paper, let $F$ be its coefficient field, and let $V$ b…

Local freeness conjecture. For every $(\varphi,\Gamma_K)$-module $M$ over $\mathcal{R}_A(\pi_K)$, there exists a finite admissible cover $\{\operatorname{Max}(A_i)\}_{i=1,\dots,n}$…

Extension conjecture. The claim of the theorem and the corollary also holds true if one replaces $x_0$ by a (locally) closed subscheme of the special fiber over which there exists…