4 problems

Localisation injectivity and vanishing conjecture. For all $0\leq i\leq n$, this map is injective, and

Kim's conjecture. For $n\gg 0$,

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_i/F$ be quadratic extensions in which $\mathfrak p$ is inert, wit…

Arithmetic ring-specific Oort conjecture. There exist $l>0$ and a normal, $C$-Galois, regular branched cover $Y\rightarrow D$ such that $Y_k$ is reduced, $L_m=M$ for $m\gg0$ and…