241 problems

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…

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…

A $p$-adic representation that is de Rham should become log-crystalline after a finite base change. Fontaine's conjecture. De Rham representations are potentially log-crystalline.…

Let $F$ be a totally real field, let $p$ be a prime, and let $\rho$ be a mod $p$ Galois representation. For a regular weight $(k,l)$, geometric modularity means that $\rho$ is modu…

Let $K$ be a finite extension of ${\bf Q}_p$, let $G_K$ be its absolute Galois group, and let $X$ be a smooth and proper algebraic variety over $K$. Write $H^i_{\rm dR}(X)$ for de…

Rapoport–Zink's conjecture. There exists an open subspace

Let $p$ be an odd prime, let $k$ be even, and let $a_p$ be the relevant crystalline Frobenius trace. Write $\overline V_{k,a_p}$ for the reduction modulo $p$ of the associated two-…

Let $F$) be a totally real field in which $p$ is inert, let $E/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let…

Fontaine's conjecture. Every weakly admissible filtered isocrystal is admissible.

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…

Geometric Sen perfectoidness conjecture. The object $X_\infty$ of $X_{\mathrm{pro\acute{e}t}}$ is perfectoid if and only if $\kappa_{X_\infty}^*$ is surjective, equivalently if and…

Fargues–Rapoport's conjecture. There exists a group-theoretic condition which expresses precisely when

Let $A^h$ be a local chart arising from the Bambozzi–Kremnizer spectrum of a Banach algebra over $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$, and let $S$ be the coefficient ring. For int…

Let $K$ be the underlying $p$-adic field, let $F$ be the coefficient field of Hyodo–Kato cohomology, and let $X$ be a proper, log-smooth, fine and saturated…

A potentially crystalline representation is a representation of a local Galois group that becomes crystalline after a finite extension. A potentially diagonalizable representation…

Solubility conjecture. Every small ${\mathscr O}_{\mathfrak X}[\frac 1 p]$-Higgs bundle with coefficients in $\xi^{-1}\Omega^1_{{\mathfrak X}/{\mathscr S}}$ is soluble. This is par…

Hartl's conjecture.

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…

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…

Fontaine's conjecture. Every de Rham representation of $G_K$ is potentially semistable.

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

$C_{\mathrm{EP}}(K,V)$.

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