9 problems

Let $k$ be a finitely generated field, let $\overline{k}$ be a separable closure, let $\ell$ be different from the characteristic of $k$, let $\Lambda$ be a finite extension of…

Finite-order determinant conjecture. If ${\mathcal L}_\ell$ has determinant of finite order, then for every closed point $x\in X({\mathbb F}_{q^n})$ this characteristic polynomial…

Let $\mathbb{F}_q$ be a finite field, and let $Y_0$ and $Z_0$ be varieties over $\mathbb{F}_q$. A morphism $f_0:Z_0\to Y_0$ is $\lambda$-uniform if, for every untwisted compatible…

Let $X$ be a normal and complete variety over a finite field $k$, let $U\subset X$ be a smooth quasi-projective open subvariety whose complement is the support of an effective Cart…

Let $C_0$ be a smooth projective curve over the finite field $\mathbb F_q$ of characteristic $p>0$, let $C$ be its base change to $\overline{\mathbb F}_q$, and let $M_K$ and…

Let $Y$ be the variety appearing in the definition of companions. For an abstract field isomorphism $\tau$ between the relevant algebraic closures of coefficient fields, a $\tau$-c…

Let $\ell$ be a rational prime. Let $X$ be an irreducible regular scheme, flat and of finite type over $\mathbb{Z}[\ell^{-1}]$. Let $E$ be a finite extension of $\mathbb{Q}$, let…

Let $p$ and $\ell$ be distinct primes. Let $X$ be a connected normal scheme of finite type over $\mathbb{F}_p$, and let $\mathcal{E}$ be an irreducible lisse…

Let $U$ be a smooth variety over a finite field. Consider a compatible system of lisse $\overline{\mathbb{Q}}_\ell$-sheaves on the integral closed curves contained in $U$, satisfyi…