24 problems

Let $c1$ be the representation occurring in the definition of the sheaf $c4_n^{-1}(V-B_i)^{c1}$, and let $a3_n$ be the corresponding covering of the Drinfeld half-plane. For an ope…

Non-vanishing conjecture for $B_{\xi,\zeta}$. The Banach space $B_{\xi,\zeta}$ is non-zero. This extends the preceding Satake and unramified Langlands-functoriality construction to…

Let $G$ be the group in the paper, let $\rho$ be an irreducible algebraic $\mathbb{Q}_p$-rational representation of $G$ over $K$, and let $\pi$ be a generalized Steinberg represent…

Let $V$ be a crystalline representation of ${\rm G}_{{\mathbb Q}_p}$, let $T$ be a Galois-invariant lattice, and write $V\bmod p$ for the semisimplification of $T/pT$. Let $\Pi(V)$…

Let $x:\mathT^S(K^p)[1/p]\to\overline{\mathbf Q}_p$ be a continuous ring homomorphism, let $\mathfrak m_x=\ker(x)$, and suppose that the global Langlands program supplies a corresp…

Let $K$ be a finite extension of $\mathbb{Q}_p$. Write $\mathcal{D}(\ast/GL_n(K)^{\la})$ for the stable $\infty$-category of locally analytic representations of $GL_n(K)$, let…

Let $L$ be the local field appearing in the construction, let $E$ be a finite extension of $L$ containing all embeddings of $L$ into $\overline{\mathbb{Q}}_p$, and let … be a two-d…

Let $G$ be a reductive group over $\mathbb{Q}$ admitting a Shimura datum $(G,X)$, and let $(\operatorname{Sh}_{K,\mathbb{C}})_K$ be the associated inverse system of Shimura varieti…

Generalized Emerton conjecture. For every finite set $S$, there is a factorization

Emerton's conjecture. If $\rho$ is unramified outside finitely many places, then

Let $K/{\mathbb Q}_p$ be unramified, let $\overline\rho$ be absolutely irreducible, let $\lambda=(0,0)$ for each embedding, and let $t$ be a tame inertial type of level $f$. Let…

Let $E\subset\overline{\mathbb{Q}_p}$ be a finite extension of $\mathbb{Q}_p$, and let $\rho:G_{\mathbb{Q}}\to\operatorname{GL}_2(E)$ be an odd continuous absolutely irreducible re…

Let $F$ be a finite extension of ${\mathbb Q}_p$, let $E$ be a sufficiently large coefficient field, and write $G=GL_n(F)$. Let $\pi$ be an absolutely irreducible generic represent…

Let $\rho$, $\Sigma_p$, $C(\rho)$, $\Sigma(\rho)$, $\mathcal E(U^p)_{\overline\rho}$, and $\delta_J^c$ be as in the preceding setup. Partial classicality conjecture. For every subs…

Let $T_p$ be the relevant diagonal torus, let $\mathcal E(U^p)_{\overline\rho}$ be the eigenvariety attached to the definite unitary group, and retain the notation for…

Breuil's conjecture. For a continuous character $\chi$ of $T_p$ over $E$,

Breuil's conjecture. For a character $\chi:T(L)\rightarrow E^{\times}$,

Let $\rho$ be an $n$-dimensional continuous representation of $\operatorname{Gal}(\overline{F}/F)$ over $E$, with $\rho_{\tilde v}$ crystalline and very regular for every…

Let $\rho$ be the Galois representation attached to an automorphic representation of a unitary group of semisimple rank $2$, and let $\tau_v\subset\pi_v$ be the $K$-type correspond…

Let $X$ be the parameter space appearing in the global setting, let $L$ be the coefficient field, and let $\Sigma_p$ be the set of places above $p$. For a generic ordinary point…

Let $\rho_p:G_{\mathbb{Q}_p}\rightarrow\operatorname{GL}_2(E)$ be a continuous de Rham Galois representation, and let $B(\rho_p)$ denote the associated $p$-adic local Langlands rep…

Let $B(\rho_x)^{\circ}$ and $\mathcal{B}_x^{\circ}$ be the lattices in the $p$-adic constructions, let $M^{\circ}$ be the specified lattice in…

Let $\mathscr{S}_*^{\mathrm{cris}}$ denote the crystalline parameter space used in the paper. For $s\in\mathscr{S}_*^{\mathrm{cris}}$ non-exceptional, let…

Let $L$ be a finite extension of ${\mathbb Q}_p$, let $\mathscr{S}_{\mathrm{irr}}$ parameterize the two-dimensional irreducible trianguline representations, and for…