5 problems

Let $\mathfrak{q}$ be a prime ideal with norm $q$, and suppose a class in the relevant finite-flat extension group is unramified outside $\mathfrak{q}$. Phantom-class criterion. Th…

Let $\mathfrak{q}$ be a prime ideal with $N(\mathfrak{q})=q\not\equiv1\pmod p$. Suppose that the finite-flat extension group $\operatorname{Ext}^1(\mu_p,\mathbf{Z}/p\mathbf{Z})$ ov…

Kummer group scheme classification conjecture. Any model of $\mu_{p^n,K}$ over ${\mathcal O}_K$ is a Kummer group scheme.

Let $k$ be a perfect field of characteristic $p$, let $W(k)$ be its ring of Witt vectors, let $K_0=W(k)[1/p]$, and let $K$ be a finite totally ramified extension of $K_0$ with ring…

Higher-level canonical subgroup conjecture. The canonical subgroup of higher level also appears in the Abbes–Saito filtration of $G[p^n]$.