23 problems

Deligne's weight-monodromy conjecture. The action of Frobenius on $\operatorname{gr}^M_j H^i(Y_{\overline{K}},\mathbb{Q}_\ell)$ should be pure of weight $i+j$.

Let $K$ be a nonarchimedean local field, let $X/K$ be a proper smooth variety, and let $0\leq i\leq 2\dim X$. Write $H_{\text{\'et}}^i(X_{\bar K},\Q_\ell)(\frac i2)$ for the $i$-th…

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

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…

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

Let $K$ be a complete discrete valuation field with residue field $k$, let $\ell$ be different from the characteristic of $k$, and let $X$ be a proper smooth scheme over $K$. For…

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…

Weight-monodromy conjecture. The weight filtration coincides, up to a shift, with the monodromy filtration. Consequently, the Clemens--Schmid sequence is exact.

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…

Let $K$ be a local field with residue characteristic $p$, let $Y/K$ be a smooth connected variety with basepoint $y\in Y(K)$, and let $U/\mathbb Q_\ell$, for $\ell\neq p$, be the…

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 the smooth projective variety and $H^i_l(X)$ the l-adic cohomology considered in the source. Let $M_*H^i_l(X)$ be its monodromy filtration, and write…

Let $F$) be a local field of residue characteristic $p>0$, let $X$ be a smooth and proper variety over $F$, and let $\ell$ be a prime different from $p$. Suppose that $X$ has semis…

Let $X$ be a projective smooth variety over a finite extension $K$ of $[?]$. For any prime $p\neq \ell$ and any integer $i\geq 0$, consider the $\operatorname{Gal}(\overline K/K)$-…

Let $Y$ be the reduction of a projective semistable $A$-scheme as in the source, and consider its weight spectral sequence … This induces the weight filtration on…

Let $X/\mathbf Q$ be a smooth proper variety, and let $H_p^i(X)$ be its $p$-adic étale cohomology. The Frobenius acts on…

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…