41 problems

Let $F$ be a totally real field, let $p$ be unramified in $F$, and let $\rho:\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ be a continu…

Let $\rho$ be a continuous, odd, irreducible representation … and let $W(\rho)$ be the set of Serre weights predicted by the Buzzard–Diamond–Jarvis recipe. A Serre weight is an irr…

Buzzard–Diamond–Jarvis conjecture. The representation $\rho$ arises from a Hilbert modular form. This is the folklore modularity conjecture underlying the Serre weight conjecture;…

Herzig's Serre weight conjecture. The set $W^{?}(\rhobar|_{I_{\Qp}})$ should give the set of regular modular weights.

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 $\rho$ be a crystalline representation with Hodge--Tate weights $\boldsymbol{\mu}=(\text{undefined})$, and let $\tau(\boldsymbol{\text{undefined}},w)$ denote the corresponding…

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 $\rho:\operatorname{Gal}(\overline{F}/F)\rightarrow\operatorname{GL}_2(\overline{\mathbb{F}}_p)$ be a continuous, totally odd, irreducible representation for a totally real fie…

Let $\overline{\rho}:G_K\to\mathrm{GL}_n(\mathbb{F})$ be a tame continuous homomorphism, and let $W^{?}(\overline{\rho})$ be its predicted set of regular Serre weights. A globaliza…

Let $F/F^+$, $G$, $U$, $S(U,V)$, and the Hecke maximal ideal $\mathfrak{m}$ be as above, with associated absolutely irreducible representation…

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 $\overline{r}$ be a global Galois representation in the setting of the paper, and let its refined Serre weights be determined by its restriction at $p$. The weight part of Serr…

Let $\overline\rho:G_K\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ be an irreducible continuous representation, let $\lambda=(0,0)$ for each embedding, and let ${\mathcal D}$ be…

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…

Assume $F={\mathbb Q}_{p^f}$, the Hodge type is $\lambda=0$, and ${\rm t}$ is tame. Let $\overline\rho$ be the residual representation, let $R^{\psi}({\rm t},\overline\rho)$ be its…

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 $F$ be a totally real field, let $p$ be a prime, and let $\rho:G_F\to\mathrm{GL}_2(\overline{\mathbf{F}}_p)$ be a representation. For each prime $\mathfrak{p}\mid p$ of $F$, wr…

Let $\bar{\rho}:G_K\to\operatorname{GL}_d(\mathbf{F})$ be a residual representation, let $W(\bar{\rho})$ be its conjectural set of Serre weights, and let $Z_{\boldsymbol{k}}$ be th…

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