38 problems

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 $G$ 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…

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…

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 $(r,N,V)$ be a Weil–Deligne representation together with a filtration type, and let the associated locally algebraic representation of a general linear group be the representat…

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 $f$ be an eigenform of weight $k\geq 2$ on $\Gamma_1(N)$ for some $N$ prime to $p$, and let $\rho_f\!\mid_{{\rm G}_{{\mathbb Q}_p}}$ be the associated $p$-adic representation o…

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 $G/\mathbf Q$ be a connected reductive group, let $K^p\subset G(\mathbb A_{\mathbf Q}^{p,\infty})$ be a compact open subgroup, and let $L/\mathbf Q_p$ be a finite extension. Wr…

Let $p>2$, let $K=\mathbf Q_{p^f}$, let $\overline\rho$ be a $2$-dimensional mod $p$ Galois representation, and let $R_\infty$ be a patched deformation ring with a formally smooth…

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…

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 $X_{\mathfrak p}(\overline\rho)$ be the relevant trianguline eigenvariety, let $y=(r_y,\delta)$ be the point under consideration, and let $\chi_J^c$ be the characters defined f…

Let $\widehat{T}_L^{\mathrm{spl}}$ be the subspace of characters whose adjacent ratios are special, let $K_0({\lambda}_{\mathbf h},\underline{\delta}_{\mathrm{sm}})$ be the free ab…

Emerton–Gee–Hellmann's locally analytic categorical Langlands conjecture. The functor $\mathfrak{LL}_p^{la}$ is fully faithful when restricted to a suitable category of “tempered”…

Let $E$ be a finite extension of $\mathbb{Q}_p$ with coefficient field $C$, let $\rho_p:G_{\mathbb{Q}_p}\to\mathrm{GL}_2(E)$ be a de Rham representation with Hodge–Tate weights…

Let $V$ be irreducible, crystalline, Frobenius semisimple with distinct Hodge–Tate weights $0$ and $k-1$. Let $\breve{\alpha}$ and $\breve{\beta}$ be the relevant Frobenius eigenva…

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 $x$ be a closed $E$-valued point of the relevant deformation space, let $r_x$ be the Galois representation associated with $x$, and let $V(r_x)$ be the admissible unitary $E$-B…

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…

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

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