28 problems

Termination conjecture for the motivic filtration. The filtration terminates, meaning

Let $X$ be a scheme of finite type over $\mathbb{Z}$, let $d=\dim X$, and assume that higher Chow groups $CH^r(X,i)$ are defined for such schemes, have finite ranks, and satisfy…

The algebraicity conjecture. For all integers $b\in\mathbb N^*$ and $p\in\mathbb N$ there is an integer $D$ depending only on $b$, $p$, $Y$ and ${\mathcal O}(1)$, such that for all…

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

Let $F$ be a field. Let $B_2(F)$ be the Bloch group, let $R_3(F)$ be the subgroup generated by the stated relations in ${\mathbb Z}[{\mathbb P}^1_F]$, and define … Let…

Let $F$ be a field. For $m\ge 3$, let ${\mathcal G}_m(F)$ be Goncharov's motivic complex group in degree one, let ${\delta}_m$ be its differential, and define … Write…

Toric–Sreekantan regulator conjecture. For each prime $\ell$, the valuation of the toric regulator at $\ell$ is the Sreekantan regulator tensored with ${\mathbb Q}_\ell$. This conj…

Beilinson's conjecture for higher Chow groups. This map should be injective.

Let $X$ be a smooth projective variety, and let $\underline{CH}^{k}(X,m)$ be the relevant higher Chow group, $\underline{r}_{k,m}$ its regulator, and $D(k,m)$ the indicated denomin…

Bloch–Beilinson filtration conjecture. There exists a filtration

Let $E_1,E_2,\ldots,E_n$ be very general complex elliptic curves and let … For $1\leq m\leq n$, define … The real regulator is the map $r_{k,1}$ on…

Pseudo-localization conjecture. The respective cokernels $\mathcal{C}_{\bullet}$ and $\mathcal{C}'_{\bullet}$ of these two restriction morphisms are acyclic.

Graph isomorphism conjecture. For every such $A$, the graph homomorphism is an isomorphism.

Let $\mathcal{X}_{\mathcal{U}}$ be the total space of the family over $\mathcal{U}$, let $\mathcal{H}^{2p-r}_v(p)$ be the varying part of the relevant polarized variation of Hodge…

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…

Hodge-$\mathcal D$-Conjecture. The map $r_{\mathcal{D},\mathbb{R}}^{p,n}$ is surjective. This is a higher-dimensional and higher-cycle analogue of the Hodge conjecture, formulated…

Let $W$ be a smooth projective variety over $\overline{\mathbb{Q}}$, and let $r$ and $m$ be integers for which the displayed higher Chow group is defined. Let … be the generalized…

For a number field $k$, let $\mathrm{CH}^n(k,2n-1)_{\mathbb{Q}}$ be the simplicial higher Chow group and let $\mathrm{LCH}^n(k,2n-1)_{\mathbb{Q}}$ be the image of the canonical hom…

Let $\mathcal{O}_K$ be an excellent henselian discrete valuation ring with quotient field $K$ and residue field $k=\mathcal{O}_K/\pi\mathcal{O}_K$, assume that $1/n\in k$, and let…

Let $k$ be a finite field, let $\overline{k}$ be a separable closure, and let $G_k$ be its Galois group. Let $X$ be a projective smooth variety over $k$, let $\ell$ be a prime diff…

Integral $d\log$ conjecture. The map $d\log_m$ is surjective. The rationalized map is the generic-point cycle class map $\lim(\textup{cl}_{m,m})$; the source notes that rational su…

Generic-point cycle class conjecture. For all $r,m\geq 0$, statement (S3) holds. The supplied span does not reproduce statement (S3), so its precise mathematical content cannot be…

Let $X$ be smooth and projective over a finite field, and let $H_i^K(X,\mathbb Q/\mathbb Z(1))$ denote Kato homology in weight $1$. Write…

The non-Archimedean Hodge–${\mathcal D}$-conjecture. The map

Let $K$ be a field satisfying the rank conjecture of Suslin, and let $CH^p(K,p+q)$ denote its cubical higher Chow groups, where $q\geq 0$. A fractional linear generation conjecture…