144 problems

Let $k$ be a field and let $w\geq 0$. The group … denotes étale hypercohomology with $p$-local motivic coefficients. Beilinson–Lichtenbaum conjecture. One has … This is presented a…

Let $F$ be a field. Write $\mathcal{A}^{\star,\star}_{F}$ for the algebra of motivic Steenrod operations and $H\mathbb{F}_{p,F}^{\star,\star}H\mathbb{F}_{p,F}$ for the correspondin…

Bloch–Beilinson filtration conjecture. There exists a filtration

Beilinson's conjecture. One has

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 $F$ be a field, and let $K^{(q)}_{2q}(F)$ and $\operatorname{CH}^q(\operatorname{Spec}F,2q)$ have the meanings used in the Beilinson–Soulé vanishing conjecture. Beilinson–Soulé…

Pour $n\geq 1$, soit $\iota^\bullet(n):\mathcal{B}^\bullet(F,n)\rightarrow\mathrm{CE}^\bullet(\mathcal{C}(F))_n$ le morphisme de complexes induced by motivic polylogarithms. Conjec…

Let $A^j_{\eta}$ be the $j$-fold power of the universal elliptic-curve family over the generic point $\eta$, and let…

Simpson's motivicity conjecture. Rigid integrable connections are motivic.

Goncharov's conjecture. For any field $\mathbb K$ there is a functorial isomorphism

Let $S$ be a derived scheme. Write $\mathrm{H}\mathbb Q$ for the rational motivic cohomology spectrum over $S$, and let $\mathbb{1}_{\mathbb Q}^+$ denote the positive eigenspace i…

Let ${\mathcal X}$ be a regular scheme of pure dimension $d$ proper over $\operatorname{Spec}(\mathbb Z)$, and let $n\in\mathbb Z$. Assume that ${\mathcal X}$ satisfies Assumptions…

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 $の\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$ 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