7 problems

Let $X$ be a separated scheme of finite type over a perfect field $F$, let $U\subset X$ be dense, connected, smooth, and purely $d$-dimensional, let $\ell$ be invertible in $F$, an…

Swan trace formula conjecture. Then

Let $U$ be a smooth connected scheme of dimension $d$ over a perfect field $F$, and let $\mathcal F$ be a smooth $\overline{\mathbb F}_\ell$-sheaf on $U$. Swan class conjecture. (1…

Reeder–Yu's prediction. The Swan conductor of the adjoint Galois representation associated with the Langlands parameter of $\pi_{\infty}$ should satisfy the displayed formula. This…

Strict inequality conjecture. Assume $p=2$. For any irreducible orthogonal Galois representation $\rho$ of $G$ which is not tame, one has

Let $U$ and $V$ be as in the paper, let $\mathcal F$ be the locally constant constructible sheaf under consideration, and let $\zeta_{p^\infty}$ denote the group of all $p$-power r…

Let $G$ be the finite Galois group in the notation of the paper, let $\sigma\in G$ have order a power of $p$, and let $i$ be an integer prime to $p$. The power-invariance conjectur…