138 problems

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 $(G,[b],\{\mu\})$ be a local Shimura datum over $\mathbb{Q}_p$, with local reflex field $E$, flag variety $\mathscr{F}\ell_{G,\mu}$ over $\breve{E}$, weakly admissible locus…

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…

Berger's conjecture.

Let $L/\mathbb Q_p$ be finite, let $V$ be a finite-dimensional $L$-representation of $G_{\mathbb Q_p}$, and let $T$ be a Galois-stable $\mathcal O_L$-lattice. Let $G$ be the releva…

Let $(R,T,\zeta)$ be a triple as in the paper's setup, where $T$ is an $R$-representation of $G_{\mathbb Q_p}$ and $\zeta$ is a $\mathbb Z_p$-basis of…

Let $\mathcal{X}$ be a smooth proper $W(k)$-scheme, where $k$ is a perfect field of characteristic $p>0$. Suppose that $Z_0$ is a codimension $n$ cycle in…

Nakamura's conjecture. For every such $M$, there exists an isomorphism

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…

The local companion points conjecture. The following conditions are equivalent:

Small $p$-adic Simpson comparison conjecture. The canonical morphism induces a quasi-isomorphism

Let $V$ be an $o_L$-linear representation of $G_L$, let $M_{\mathrm{LT}}=\mathbb{D}(V)$ be an étale $L$-analytic $(\varphi_L,\Gamma_L)$-module over $\mathbf{A}_L$, and let…

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $K$ be a coefficient field containing $L$, and let $V$ be an $L$-analytic $o_L$-linear representation of $G_L$. Let $M$ be the…

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…

Crystallinity conjecture. If $pi$ is unramified at $p$, then

Hartl's local-system maximality conjecture. The space $\breve{\cal{F}}^0_b$ is the largest open $\breve E$-analytic subspace of $\breve{\cal{F}}^{wa}_b$ carrying a local system…

Hartl's connectedness conjecture. The space $\breve{\cal{F}}^0_b$ is arcwise connected.

Hartl's value-group conjecture. One has

Let $F=\mathbb{Q}$ and let $X$ be a hyperbolic curve. For the étale unipotent fundamental group $U^{\mathrm{et}}$ and its level-$n$ quotient $U_n^{\mathrm{et}}$, let…

$C_{\mathrm{Iw}}(K_\infty/K,V)$. The canonical trivialization satisfies

$C_{\mathrm{EP}}(L/K,V)$. The map $\delta_{V,L/K}$ sends $\Delta_{\mathrm{EP}}(L/K,T)$ onto $\mathbf{Z}_p[G]_{V,L/K}$.

Let $\Lambda$ be the Iwasawa algebra of the relevant $p$-adic Lie group, let $\widetilde\Lambda$ be its unramified coefficient extension, and let $\mathbb T$ be the associated big…

Let $\overline{\Omega}$ be the closure of the crystalline locus in the deformation space $X(\overline{\rho})$, and let $\rho \in \overline{\Omega}$ be a classical modular point. Sa…

Mokrane's spectral sequence conjecture. For all $r,w$, the map

The $p$-adic weight-monodromy conjecture. The filtrations satisfy