19 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…

Let $L/\mathbb{Q}_p$ be a finite extension, let $E$ be a coefficient field, let $W_L$ be the set of Serre weights, and let $\overline{\rho}_L:G_L\to\mathrm{GL}_2(\mathbb{F})$ be a…

Let $G$ be an unramified group over $\mathbf{Q}_p$ satisfying Hypothesis, and let $d=\dim\check{\mathcal{B}}$ be the dimension of the flag variety of the dual group. For each Serre…

Let $G=\operatorname{GL}_n$. For each Serre weight $\sigma$ of $G(\mathbf{F}_p)$, let $\mathcal{Z}(\sigma)$ be a cycle in the $d$-dimensional cycle group of the reduced Emerton–Gee…

Let $\sigma$ range over parahoric Serre weights, and let $\mathcal{X}^{\operatorname{crys},\lambda,\tau}_{{^{L}\!G},\mathbb{F}_p}$ have cycle class in the Chow group of…

Let $\lambda$ be a Hodge type, viewed as an element of the character lattice of $G$, and let $\tau$ be a tame inertial type. Let $V(\lambda-\eta_{\mathbb{Q}_p})$ denote the restric…

Let $\overline r$ be as in the paper, and let $W_{\mathrm{gen}}(\overline r)$ be the subset of generic modular Serre weights. Let $W^{\mathrm{BM}}_{\mathrm{gen}}(\overline r_p)$ be…

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 ${\mathcal X}_{{\mathrm{Sym}},K,\mathrm{red}}$ be the reduced Emerton--Gee stack, and let ${\mathcal Z}_{\lambda,\tau}$ be the $4f$-dimensional cycle attached to a regular Hodg…

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 $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}_K$ and residue field $k$, let $G_K$ be its absolute Galois group, and let…

Let $\boldsymbol{\rho}$ range over residual representations and let $\boldsymbol{\lambda}$ range over regular tuples of labeled Hodge–Tate weights. For each Serre weight…

Let $K$ be a non-archimedean local field of residue characteristic $l$, let $\mathbb{F}$ be a finite field of characteristic $p\ne l$, and let…

Let $K$ be a non-archimedean local field of residue characteristic $l$, let $\mathbb{F}$ be a finite field of characteristic $p\ne l$, and let…

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…

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…