27 problems

Let $K$ be the local field, let $X$ be a proper smooth variety over $K$, and let $H^j(X^{\mathrm{ca}},\overline{\mathbb{Q}}_l)$ carry its monodromy filtration $M_\bullet$ and weigh…

Let $U_L^1/U_L^h$ be a quotient of unit groups, let $\chi\colon U_L^1/U_L^h\to \overline{\mathbb{Q}}_\ell^\times$ be a character, and let $\mathbb{U}_{h,k}(\mathbb{F}_{q^n})$ be th…

Let $f:Y'\to Y”$ be a proper map between schemes over an algebraically closed field, and let $\CF$ be an irreducible $\ell$-adic perverse sheaf on $Y'$ obtained by the six operatio…

Kahn's conjecture. For every prime $l$ and any $i$, the map $c$ induces 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…

Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let ${\mathbf{L}}=\{{\mathcal{L}}_\lambda\}$ be an $E$-compatible system of liss…

Log $l$-adic relative monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\overset{\circ}{S}$, then $P$ is the monodromy filtration of $N$ relative to…

Log $l$-adic relative monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\kappa$, then the relative monodromy filtration on the cohomology relative to…

Kato's log $l$-adic monodromy-weight conjecture. If $\overset{\circ}{X}$ is projective over $\kappa$, then, for every $e\in{\mathbb Z}_{\geq 1}$, $N$ induces an isomorphism

Slope-bound conjecture. For all $i>d$, the $q$-slopes of these eigenvalues are contained in

Monodromy-weight conjecture. For every integer $m$, $R^mf_*(\overline{\mathbb{Q}}_{\ell})$ is an object of ${\mathcal A}_Y$ of pure $W$-weight $m$.

Let $X$ be a smooth projective variety over a finitely generated field $F$. Let ${\mathrm{Ch}}^r(X)$ be the Chow group of codimension-$r$ algebraic cycles on $X$ defined over $F$ m…

Drinfeld stratification conjecture. If $R_{0+}\subset R(\mathbf{S},\mathbf{L})$ (or, equivalently, $\mathbf{G}^{0}(\mathbf{S},\theta)\subset\mathbf{L}$), then

Let $X$ be a smooth projective variety over the finitely generated field $K$, and let $CH^i_l(X)$ denote the subgroup of cycles homologically equivalent to zero for $l$-adic cohomo…

Let $k$ be a number field and let $f:X\to Y$ be a smooth proper morphism over $k$. Fix $p,q\in\mathbb{Z}_{\geq 0}$, and for each prime $\ell$ let…

Let $X_h$ be a finite-type variety over a finite field with geometric Frobenius $\Fr_{q^n}$ acting on its compactly supported cohomology. A variety is maximal in the sense of Boyar…

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 $K$ be a global field, let $X$ be a smooth projective $K$-variety of dimension $n$, and fix $i\in[0,2n]$. For a finite place $v$ and a prime $l$ not dividing the residue charac…

Let $X$ be a scheme over $\mathbf{F}_p$, and let $F$ be a constructible $\mathbf{Q}_l$-sheaf on $X$. Purity conjecture. Every constructible $\mathbf{Q}_l$-sheaf on $X$ is mixed. Th…

Let $U$ be a non-empty Zariski-open sub-curve of the curve under consideration. Let $\mathfrak A$ and $\mathfrak B$ be abelian schemes over $U$, and let $A$ and $B$ denote their ba…

Let $k=\mathbb F_q$ be a finite field with $q=p^f$, let $L$ be the function field under consideration, let $X$ be a smooth projective variety over $L$, let $r$ be a codimension, an…

Let $\mathbf{F}=\mathbf{F}_q$ be a finite field, let $X$ be a smooth variety over $\mathbf{F}$, let $\bar X=X\times_{\mathbf{F}}\overline{\mathbf{F}}$, and let…

Let $\mathbf{F}=\mathbf{F}_q$ be a finite field, let $X$ be a smooth projective variety over $\mathbf{F}$, let $\bar X=X\times_{\mathbf{F}}\overline{\mathbf{F}}$, and let…

Let $K$ be a non-archimedean local field with finite residue field $k$, let $X$ be a smooth projective variety over $K$, and let $M_\cdot$ be the monodromy filtration on…

Primitive-character cohomology conjecture. As a virtual module for $\operatorname{Gal}(\overline{\mathbf F}_q/{\mathbf F}_{q^h})$, one has $\dim\mathcal{H}_\psi=(-1)^{h-1}$, and th…