206 problems

Tate conjecture. The representation $\rho_{\ell}$ is semisimple, and the cycle class map

Let $X$ be an arithmetic scheme, meaning a separated scheme of finite type over $\operatorname{Spec}\mathbb{Z}$, and let $n<0$. Write $X_{\text{ét}}$ for its étale site and…

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…

Lyubeznik's conjecture.

Let $X$ be a closed subscheme of a smooth scheme over a perfect field $k$, and let $\mathcal{F}$ be a constructible complex of $\Lambda$-modules of finite Tor-amplitude on $X$. Let…

Let $k$ be the algebraic closure of a finite field $\mathbb{F}_q$, and let … be a correspondence, with $X$, $C$, $c_1$, and $c_2$ defined over $\mathbb{F}_q$, and let $\mathcal{F}$…

Jannsen's generalized Tate and Beilinson conjectures. For all $i,j\in{\mathbb Z}$, the regulator map induces an isomorphism

Grothendieck–Serre semi-simplicity conjecture. This action should be semi-simple; equivalently, the Frobenius action on $H^{i}(\bar X,\mathbb Q_\ell(j))$ should be diagonalizable o…

Let $E$ be a $p$-adic field, let $F$ be the coefficient field defining the Fargues–Fontaine curve, and let $X^{\operatorname{alg}}_{E,F}$ denote its algebraic version. Let…

Fontaine's ramification conjecture. The group $G_K^{(\mu)}$ should act trivially on $\mathrm{H}^i_{\text{\'e}t}(X_{\overline{K}}, \mathbb{Z}/p^n\mathbb{Z})$ when

Let $\mathcal S$ be a henselian trait with perfect residue field of characteristic $p>0$, with closed point $s$, generic point $\eta$, and geometric generic point $\overline\eta$.…

Let $X$ be a proper regular arithmetic scheme of dimension $d$, let $n<0$, and let $H^i(X_{\text{\it \'{e}t}},\mathbb{Z}^c(n))$ denote negative-weight motivic cohomology. Geisser’s…

Coefficient comparison conjecture. There are equivalences of triangulated categories

Weil's Riemann hypothesis conjecture. The algebraic numbers appearing as Frobenius eigenvalues satisfy

Suppose that $k$ is a field and that $ell$ is a prime distinct from the characteristic of $k$. Let $K/\ell$ be the mod $\ell$ algebraic $K$-theory presheaf of spectra on smooth $k$…

Let $K$ be a global field, let $X/K$ be a smooth projective geometrically integral variety of dimension $d$, let $n$ be prime to the characteristic of $K$, and let $i>0$. Set…

Perfectness conjecture. The pairing is conjectured to be perfect.

Friedlander's generalized isomorphism conjecture. This map should be an isomorphism. The conjecture is used in the source to relate étale cohomology to the cohomology of discrete g…

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…

Let $X$ be a variety in the generality considered in the source, with etale motivic cohomology with compact supports and etale motivic Borel–Moore homology related by the natural p…

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…