138 problems

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 $L/\mathbb Q_p$ 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 $F=\mathbb{Q}$ 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

Perrin-Riou's integrality conjecture. One has

La conjecture $C_{EP,F}(V)$. Pour tout $\omega\in\Delta_F(V)$, on a

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…

Reductive-group triangulation uniqueness conjecture. Then $D$ has at most one triangulation of parameter $\delta$.

For an integer $n\geq 2$, let $R\Gamma(\mathbb{C}_p^\mathrm{Syn},\mathcal{O}\{n\})$ denote analytic syntomic cohomology of $\mathbb{C}_p$ with coefficients in the $n$-th Tate-twist…

Let $X$ be a smooth rigid-analytic space over a $p$-adic field $K$, and let $\mathbb{L}$ be a de Rham $\mathbb{Q}_p$-local system on $X$. A point $x\in X$ has rank $1$ if it is a r…

Let $L/\mathbb{Q}_p$ be a $p$-adic field and let $X/L$ be a smooth rigid analytic variety. Let $\mathrm{AdmPair}^{\mathrm{basic},\mathrm{pgr}}(X^\diamond)$ denote basic admissible…

Potentially unramified tower conjecture. The following assertions hold: (1) $\hat{\tilde{X}}$ is an adic space and $(\hat{\tilde{X}})^\diamond=\tilde{X}^\diamond$; (2) the natural…