17 problems

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $\ell$ and $\ell'$ be distinct primes with $\ell,\ell'\neq p$, and let $X$ be a normal variety over $\mathbb{F}_q$.…

Deligne's Weil II conjecture. The sheaf $V$ has weight $0$; there is a number field $E(V)\subset\bar{\mathbb{Q}}_\ell$ containing all coefficients of the local characteristic polyn…

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 $k$ be algebraically closed, let $r,s\geq 1$, and let $p$ be relatively prime to $r$ and $s$. Let … be a formal Laurent series in $k((1/t))$ with $a_{-s}\neq 0$, and let…

Let $k$ be algebraically closed. For a Laurent series … with $a_{-s}\neq 0$, define its formal Legendre transformation $L_f(t')$ by … Write $L_f(t')$ as a Laurent series in…

Mixed realization conjecture. The functor $\mathop{\rm real}\nolimits_\Delta$ should factor through a mixed realization functor sending ${\mathbf D}(\Delta)$ to mixed Hodge modules…

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 an irreducible smooth variety, let $l$ be a prime different from the characteristic, and let $L$ be an irreducible lisse $\overline{\mathbb{Q}}_{l}$-sheaf…

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 $k$ be a perfect field, let $X/k$ be a normal variety, and let $j:U\hookrightarrow X$ be a $k$-smooth open subscheme whose closed complement is the support of an effective Cart…

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…

Extension conjecture. The formula remains true for arbitrary semisimple $K$ and $L$. The displayed formula is proved in the stated setting where $K$ and $L$ arise from pure pervers…

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…

Let $X$ be a normal scheme of finite type over ${\mathbb{F}}_p$, let $\ell\ne p$ be a prime, and let $\mathcal{E}$ be an irreducible lisse $\overline{{\mathbb{Q}}}_{\ell}$-sheaf on…