7 problems

Let $M$ be a motive in the chosen motivic category, let $G_{l,K}^{\operatorname{alg}}$ be the algebraic group attached to its $l$-adic realization, and let…

Let $(V,\psi)$ be a rational polarized Hodge structure and let $(V_l,\rho_l)$ be a family of $l$-adic representations satisfying conditions (D1), (D2), and (R1)–(R4). For every fin…

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 $n\ge 1$,…

Algebraic Sato–Tate conjecture. There is an algebraic subgroup $\operatorname{AST}(A)$ of $\operatorname{GSp}_{2g}$ over $\mathbb Q$, called the algebraic Sato–Tate group, such tha…

Let $X_0$ be a smooth geometrically connected variety over a finite field $k={\mathbb F}_q$ of characteristic $p\neq\ell$, let $X=X_0\otimes_k\bar k$, and let…

Let $X_0$ be a smooth geometrically connected variety over a finite field $k={\bb F}_q$ of characteristic $p\neq\ell$, let $X=X_0\otimes_k\bar k$, and let…

Let $k$ be a finitely generated field of characteristic $p>0$, let $X$ be a smooth geometrically connected $k$-variety equipped with a GLP $[?\rho$ and associated groups and covers…