8 problems

Let $Y$ be a scheme over $\mathbb Z$, and let $\{(Y_p,{\mathscr G}_p)\}$ be a family arising from a fixed constructible complex and morphisms as in the setup. Suppose $Y$ is an alg…

Let $X/\mathbb{F}_{q}$ be a smooth variety, and let $\mathcal{F}$ be an irreducible coefficient object on $X$ with algebraic determinant. Companions conjecture. There exists a numb…

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasi-projective variety, let $l\neq p$ be a prime, and let $L$ be a lisse $\overline{\mathbb{Q}}_{l}$-sheaf of rank…

Let $n$ be prime and let $\psi:\mathbb F_q\to\overline{\mathbb Q}_\ell^{\times}$ be nontrivial. Let $\Gamma=\varinjlim\widehat{\mathbb F_{q^n}^{\times}}$, and let…

Keep the hypotheses of Conjecture A: $k$ is perfect, $X/k$ is normal, $j:U\hookrightarrow X$ is a $k$-smooth open immersion with complement supporting an effective Cartier divisor,…

Let $C$ be a smooth curve over a finite field $\mathbb F_q$, let $\bar C$ be its smooth compactification, and let $M$ be the set of irreducible lisse $\bar{\mathbb Q}_\ell$-sheaves…

Let $k$ be a finite field of characteristic $p$, let $U$ be a smooth geometrically connected curve over $k$, and let $\mathscr{F}$ be an irreducible smooth…

Let $G^{sc}$ be the simply connected cover of the derived group of a reductive group $G$, and let $\operatorname{Bun}_{G^{sc}}$ be the stack of $G^{sc}$-torsors on $X$. A smooth…