112 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…

ℓ-adic integral Tate conjecture in codimension $c$. The cycle class map

ℓ-independent Lefschetz standard conjecture in degree $i$. For $X$, $h$, and $i$ as above, there exists a codimension $i$ cycle…

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 $\alpha^*$ be the inverse-image functor from etale to Zariski motivic complexes, let $j^*{\Bbb Z}_l(n)_S$ be the indicated arithmetic etale Tate-twist complex, and let $n\ge0$.…

Let $S$ be the arithmetic scheme in the source, let $j$ be the relevant open immersion, let $e$ be the indicated cup-product class, and let $A(1)$ be the real Tate-twist object. Co…

Let $S$ be the open arithmetic scheme and let $A(n)$ denote the real Tate-twist object used in the source. Theorem 12.1(a) gives a comparison isomorphism for positive twists betwee…

Finiteness conjecture. These groups are finite. This conjecture is introduced to obtain finite-generation consequences for motivic cohomology when combined with the etale realizati…

Zariski realization conjecture. (a) This morphism is an isomorphism. (b) The Zariski sheaf $R^{n+1}\alpha_*{\Bbb Z}_l(n)^c$ is torsion-free. Under resolution of singularities and t…

Etale realization conjecture. This morphism is an isomorphism. Equivalently, after tensoring with ${\Bbb Q}$, the induced morphism is an isomorphism. The mod-$l$ reduction is alrea…

Let ${\mathscr X}$ be the arithmetic scheme in the paper, with Zariski and small étale sites ${\mathscr X}_{{\mathrm{Zar}}}$ and ${\mathscr X}_{{\mathrm{\acute{e}t}}}$, and let…

Comparison conjecture. (1) There is an isomorphism in $D^b(X_{{\mathrm{\acute{e}t}}},{\Bbb Z}/p^r{\Bbb Z})$

Let ${\mathscr F}$ be a constructible étale sheaf of flat $\Lambda$-modules on $\eta$, and let $r>1$ be a critical slope of ${\mathscr F}$. Assume that … Thus $r$ is the unique cri…

Let $X$ be a smooth, projective scheme over ${\cal O}_K[1/S]$, and let $\bar{X}:=X\times_{{\cal O}_K[1/S]}\bar{K}$ be its base change to an algebraic closure. For integers $i$ and…

Swan trace formula conjecture. Then

Let $G$ be an algebraic group over an algebraically closed field $k$, and let $n$ be a positive integer such that the characteristic of $k$ does not divide $n$. The simplicial sche…

Beilinson's Tate conjecture. In case $k$ is finitely generated over $\mathbb Q$, $\operatorname{reg}_{U,\mathrm{et}}^q$ is surjective.

Beilinson–Lichtenbaum generalized Hilbert 90 conjecture. For every $n\ge 0$, one has

Beilinson–Lichtenbaum comparison conjecture. This map is an isomorphism for $p\le q+1$.

Let $k$ be a field and let $n\ge 0$. The motivic complex ${\bf Z}(n)$ is evaluated on $\operatorname{Spec}(k)$ in the étale topology. Beilinson–Lichtenbaum's generalized Hilbert 90…

Lyubeznik's conjecture.

Semisimplicity conjecture. The Galois action on

Let $X$ be a smooth projective variety of dimension $d$ over the algebraically closed field $\mathbbm{k}$, and let $f$ be a ($q$-)polarized endomorphism of $X$: there is an ample d…

Uniform torsion-freeness conjecture. There exist polynomials $\psi(x),\phi(x)\in\mathbb{Z}[x]$ such that

Let $X$ be a smooth projective variety over a number field $F$. Let $(s_{\mathbb A_f},s_{\mathrm{dR}})$ be an absolute Hodge class in … Let…