224 problems

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…

Let $X$ be a pure or mixed motive defined over $\overline{\mathbb{Q}}$. Write $\mathrm{periods}(X)$ for its periods and $\mathcal{G}_{\mathrm{mot}}(X)$ for its motivic Galois group…

Let $k$ be a field and $R$ a coefficient ring. Write $\operatorname{DM}_{\operatorname{gm}}^{\operatorname{eff}}(k;R)$ for geometric effective motives and…

Let $\mathfrak{g}^*$ be the homotopy Lie algebra associated with the free massless theory in dimension $d$, and let $G_{M,\mathrm{Betti}}$ be the motivic Galois group in the Betti…

Let $P$ be the algebra of periods and let $T$ be one of the corresponding torsors of isomorphisms, with the period construction giving a map from the period algebra to $T$. In the…

Let $K$ be the imaginary quadratic field and let $M_{\theta\mathbb{Q}_p}(n)$ be the $p$-adic realization of the Hecke-character motive defined in the source, with weight $w$. Let…

Tensor-functor realization conjecture. For every prime $p$ there exists a tensor functor

Two-dimensional Weil conjecture. For every endomorphism $M\in\operatorname{End}_{\mathcal C_{\mathbb F_q}}(X)$ there exist super Boltzmann data $(V_1,V_2,R)$ such that, for every f…

Algebra conjecture for motivic local systems. There exists a commutative associative unital algebra $A=A_{Y,D,N}$ in $\mathcal C_{\mathbb F_q}$ such that, for every $n\ge 1$,…

Let $m\geq 1$, and let $f$ and $g$ be Siegel modular forms of genus $2m$, of weights $k>2m$ and $l=k-2m$, with Satake parameters $(\alpha_0,\alpha_1,\ldots,\alpha_{2m})$ and…

Let $k$ be a field and let $\mathcal{M}_\sim(k)$ be the category of motives over $k$ for an adequate equivalence relation $\sim$. Let $\operatorname{Pic}(\mathcal{M}_\sim(k))$ deno…

Let $k$ be a field, let $\mathcal{M}(k)$ be the category of motives over $k$ for a fixed adequate equivalence relation, and let $L\in\mathcal{M}(k)$ be an invertible motive. Define…

Let $C$ be a smooth projective geometrically connected curve over $k$, of genus $g$, and let $G$ be a split semisimple connected algebraic group over $k$. Denote by…

Let $k$ be the base field, let $F$ be the ambient field extension, and let $G$ be its field automorphism group. Let $\mathcal A dm$ denote the category of admissible representation…

Let $\overline{A}$ and $\overline{X}$ be quasi-projective varieties over $\mathbb C$, with $\overline{A}$ smooth, and let $\overline{f}:\overline{A}\to\overline{X}$ be a projective…

Let $\rho:\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\operatorname{GL}_n\mathbf{Q}_\ell$ be a motivic Galois representation, and let $\delta_m(\rho)$ be the density rat…

Beilinson's conjecture. For every $X\in\mathcal V_k$ there exists a decreasing filtration $F^\bullet$ on $A^i(X)$ such that: (a) $F^0A^j(X)=A^j(X)$ and $F^1A^j(X)=(A^j(X))_{hom}$;…

Non-abelian Main Conjecture. The element $\delta_p(G_\infty,M,k)$ is induced by a generator

Equivariant Bloch–Kato conjecture. The Beilinson regulator induces an isomorphism $\Delta_f(G,M^\lor(1-k))\otimes {\Bbb{R}}\cong \1_{{\Bbb{R}}[G]}$, under which…

Let $\widehat G_{n-1}^n$ be the configuration space used to define the Grassmannian $n$-logarithm, and let $a_i,b_j$ be the maps on this space appearing in its functional equations…

Let $G$ be a finite graph with edge set $E(G)$, and let … where the sum runs over spanning trees $T$ of $G$. Let $V(Q_G)$ be the zero scheme of $Q_G$ in $\mathbf A^{E(G)}$, let…

Let $G$ be a finite graph with vertex set $V(G)$) and edge set $E(G)$. For each edge $e$, let $x_e$ be a variable, and define the Kirchhoff polynomial … where the sum runs over spa…

Cancellation conjecture for effective Chow motives. If

Nonabelian Hodge conjecture. The points of $NL(X_s,GL(n))$, that is, the $\mathbb Z$-variations of Hodge structure, are motivic representations on $X_s$.

Simpson's motivic conjecture. The representation $\varrho\otimes k$ is a direct summand over $k$ of the monodromy representation of a motive over $X$, that is, it comes from geomet…