21 problems

Fix an imaginary CM field $F$ in which $p$ is unramified, let $F^+$ be its maximal totally real subfield, and set … Let $({\mathbb Z}^2_+)^S_0$ consist of tuples $a$ satisfying…

Let $\text{undefined}:\text{undefined}\to{}^LG(\text{undefined})$ be a semisimple $L$-parameter. Let $W^{\text{undefined}}(\text{undefined})$ be the set of Serre weights arising fr…

Let $G(k)$ be the finite reductive group in the local setup, let $\overline{\rho}_0$ be the local residual representation, and let $\overline{r}$ be its suitable globalization. Def…

Let $K$ be the finite unramified extension of $\mathbb Q_p$ used in the paper, let $G_K$ be its absolute Galois group, and let $\Sigma$ be the corresponding set of embeddings. Let…

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 $F$, $\chi$, and $\overline r$ be as in the paper, with $\overline r$ automorphic of some weight and level. Let $W(\overline r)$ be its set of modular Serre weights, and let…

Let $F$) be a totally real field and let … be a continuous, irreducible and totally odd representation. Let $D$ be a quaternion algebra over $F$ that splits at exactly one infinite…

Let $\mathbb{X}$ be a fixed nondegenerate gene, and write its stratified embedded Kisin variety as … inside $(\mathbb{P}^1)^f$. Let $R^{\psi}({\rm t},\overline\rho)$ and…

Let $\text{SEKV}$ be the monoid of shape-stratified closed subvarieties of $(\mathbb{P}^1)^n$, for varying $n$, with multiplication given by products. For a tame inertial type…

Let $F^+$ be the relevant totally real field, let $\overline r:G_{F^+}\rightarrow\mathcal{G}_n(\mathbb{F})$ be automorphic, and let $\overline r_p$ be its associated local $L$-homo…

Let $F$ be a totally real field, let $p$ be a prime split completely in $F$, and let $\overline{\rho}:G_F\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ be a mod $p$ irreduci…

Let $\overline{r}:G_F\rightarrow\mathrm{GL}_n(\mathbf{F})$ be a continuous automorphic Galois representation with $\overline{r}|_{G_{F_w}}\cong\overline{\rho}_0$, where…

Let $F$ be a totally real field in which $p$ is unramified, let $\Sigma$ be the set of embeddings $F\to L$, and let $G_F$ denote the absolute Galois group of $F$. For a weight…

Artin–Hasse subspace conjecture. If $(V,J)\in W'(\chi_1,\chi_2)$ for some $J$, then

Strong Serre conjecture. If $\overline{\rho}$ is continuous, irreducible, and totally odd, then it is modular. Moreover, $\overline{\rho}$ is modular of weight $\sigma$ if and only…

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 $K/{\mathbb{Q}}_l$ be a finite extension, and let $\overline{\rho}:G_K\to\operatorname{GL}_2(\overline{{\mathbb{F}}}_l)$ be a continuous representation. Let…

Let $K/{\mathbb{Q}}_p$ be a finite totally ramified extension, and let ${\overline{\rho}}:G_K\to\operatorname{GL}_2({\overline{{\mathbb{F}}}_p})$ be a continuous representation. Le…

Fix an imaginary CM field $F$ with maximal totally real subfield $F^+$ such that every prime of $F^+$ above $p$ has residue field ${\mathbb{F}}_p$ and splits in $F$. Let $S$ be the…

Let $n\geq1$, let $p$ be a prime, and let $\overline\rho:G_{\mathbb Q}\to\operatorname{GL}_n(\overline{\mathbb F}_p)$ be continuous, odd, and irreducible. Let $F(a_1,\ldots,a_n)$ b…

Let $K/\mathbb{Q}_p$ be a finite extension with residue field $k_K$, absolute ramification degree $e$, and let $\overline\rho_K:G_K\to\operatorname{GL}_2(\overline{\mathbb{F}}_p)$…