144 problems

Let $Z(x,n)$ be written uniquely as … in $A[\![x-1]\!]$, with $h_n\geq 0$ and $\zeta_C^*(n)\in A$. Let $\operatorname{Cl}(\underline{A}(n))$ denote the class module of the Carlitz…

Let $F$ be a number field, let $X$ be a smooth projective variety over $F$, and let $M=h^{i-1}(X)(n)$. Write $の\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbbm{1},M)$ for the subspace…

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

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 $n\ge0$.…

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 $D^b(X_{{\mathrm{\acute{e}t}}},{\Bbb Z}/p^r{\Bbb Z})$

Converse weight-bound conjecture. If, for all $l$, the weights of $H^l\otimes\mathbb{Q}(M)$ lie between $l+a$ and $l+b$, then

Let $M$ be a $K$-motive, let $\Delta_K(M)$ be its fundamental line, and let $\vartheta_\infty:K_{\mathbb R}\otimes_K\Delta_K(M)\cong\mathbf{1}_{K_{\mathbb R}}$ be the period-regula…

Deligne–Beilinson conjecture.

Lichtenbaum–Milne conjecture. For every relevant integer $r$, as $t\to q^{-r}$,

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

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…

Let $T\mathcal M(n)$ denote the additive part of the degree-$n$ component of the indecomposable space, and let $Z_n(a)$ be a codimension-$n$ cycle on … For $a\in\mathbb A^1-\{0\}$,…

The motivic Hopf algebra ${\cal A}_{\bullet}(F)$ is built from motivic iterated integrals, and the map in the source identifies the corresponding motivic iterated-integral construc…

Let $X$ be a projective regular curve over an algebraically closed field $k$, and let $F=k(X)$. Let $\Gamma(F;n)$ and $\Gamma(k;n-1)$ be the weight-$n$ and weight-$(n-1)$ motivic c…

Goncharov's scissor-congruence realization conjecture. The map constructed in theorem 11.18.01.5(c) is an isomorphism.

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…

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…

Let $G_4(F)$ be the weight-four Grassmannian group, let $\mathcal B_4(F)$ be the corresponding polylogarithmic group, and let … be the canonical cobracket described in the construc…

Let $\widetilde A_n$ be the free abelian group generated by admissible pairs of simplices, let $\overline C_{2n}$ be the Grassmannian configuration group, and suppose the maps…