14 problems

Fix an odd prime $p$, a finite extension $E$ of ${\mathbb Q}_p$ with residue field ${\mathbf k}_E$, and a potentially crystalline representation…

Genetic decomposition conjecture. There exists a decomposition

Let ${\mathcal S}$ be a set of types, and let $\mathrm{JH}(\overline{\sigma}({\mathcal S}))$ be the union of the Jordan--Hölder factors of the reductions…

Let $G_K$ be the absolute Galois group of $K$, let $\overline{\rho}:G_K\rightarrow\mathrm{GSp}_4(\mathbf{F})$ be continuous, and let $R_{\overline{\rho}}^\square$ be its framed def…

Let $K/\mathbb{Q}_p$ be a finite extension, let $\overline{\rho}:G_K\rightarrow\operatorname{GL}_n(\mathbb{F})$ be continuous, and regard $\overline{\rho}$ as an $\mathbb{F}$-point…

Let $R_\rho^{\mathrm{ord}}$ be the ordinary deformation ring and let $H^2_{\mathcal P_{\mathrm{ord}}}(\operatorname{Gal}_{\mathbb Q,\Sigma},\operatorname{Lie}\check{\mathrm G})$ de…

Let $R_\rho^{\mathrm{ord}}$ be the ordinary deformation ring attached to the Galois representation $\rho$ and let $\Lambda_E$ be the ordinary weight algebra, with…

Let $\mathbb{F}$ be a finite field of characteristic $p$, let $K$ be a finite extension of $\mathbb{Q}_p$ with absolute Galois group $G_K$, and let $V_{\mathbb{F}}$ be an absolutel…

Let $R_E(GL_n(\mathcal{O}_F))$ and $R_{\mathbb{F}}(GL_n(\mathcal{O}_F))$ be the Grothendieck groups of finite-length representations over $E$ and $\mathbb{F}$, respectively, and le…

Let $K$ be the local field and let $G_K$ be its absolute Galois group. Let $\mathbb{F}$ be the coefficient field, let $\Gamma$ be the relevant Grothendieck group, and let…

Strong local Langlands interpolation conjecture. If $\overline{\rho}$ is semisimple, there is a unique isomorphism

Weak local Langlands interpolation conjecture. For any $\overline{\rho}$, there is a map

Let $K$ be a finite extension of $\mathbb Q_p$, let ${\bar{r}}:G_K\to\operatorname{GL}_n(\mathbb F)$ be continuous, and let $R^\square_{{\bar{r}},\lambda,\tau}$ be the potentially…

Let ${\bar{r}}:G_{\mathbb{Q}_p}\to\operatorname{GL}_2(\mathbb F)$ and the deformation rings and multiplicities $a_{m,n}$ and $a^{\operatorname{cr}}_{m,n}$ be as above. For each…