17 problems

Let $X$ be a proper smooth variety over a local field $k$, and let $V$ be its $i$th $\ell$-adic étale cohomology group. Write $N$ for the monodromy operator, let $W_\bullet$ be the…

Let $K$ be a discretely valued field with separable closure $K^{\operatorname{sep}}$, let $X$ be a variety over $K$ with semistable model, and let $E_r^{p,q}$ be its weight spectra…

Let $K$ be the local field and $F$ its residue field, let $X_K$ be a proper smooth variety over $K$, and let … Let $M$ be the monodromy filtration on $V$ and $W$ the weight filtrat…

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

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

Rapoport–Zink spectral sequence conjecture. For all $r,w$, the induced map on $E_2$-terms is an isomorphism

Weight-monodromy conjecture. The two filtrations agree up to the shift specified by

Let $V$ be a representation of the Langlands dual group, and let $\scrZ^{\mathrm{mix}}(V)$ be the associated mixed perverse sheaf obtained from the mixed central functor. Its monod…

Let $K$ be a local field with residue characteristic $p$, let $\ell\neq p$ be a prime number, and let $\mathcal X$ be an admissible formal $\mathcal O_K$-scheme with smooth generic…

Let $K$ be a $p$-adic local field with residue field of cardinality $q$, let $X$ be a smooth proper rigid-analytic $K$-variety, let $\ell\neq p$ be a prime number, and suppose that…

Let $X$ be a proper smooth algebraic variety over $\mathbb{C}_p$ defined over $\bar{\mathbb{Q}}_p$. Its Hyodo–Kato cohomology $H^i_{\mathrm{HK}}(X^{\mathrm{an}})$ is a…

Let $Y$ be a proper smooth algebraic variety over $K$, and let $H^i_{\mathrm{HK}}(Y_{\bar{K}})$ denote its Hyodo–Kato cohomology, equipped with Frobenius, monodromy, and the monodr…

Torsion analogue of the weight-monodromy conjecture. For every integer $i$, there exists a non-zero monic polynomial $P_i(T)\in\mathbb Z[T]$ such that its roots have complex absolu…

Weight-monodromy and independence-of-ell conjecture. The polynomial

Weight-monodromy conjecture. This representation is pure of weight $w$; namely, the eigenvalues of $\sigma(v)$ on the $r$-th graded piece of the monodromy filtration are Weil numbe…

Let $X$ be a proper, smooth, separated scheme of finite type over a non-Archimedean local field $K$, let $l$ be a prime different from the residue characteristic, and let…

Let $K$ be a perfectoid field with tilt $K^\flat$, let $X\subset \mathbb{P}^n_K$ be a smooth closed subscheme, and let $\tilde{X}\subset (\mathbb{P}^n_K)^\mathrm{ad}$ be a small op…