8 problems

Let $Z$ be a smooth irreducible variety over $k$, and let $\varphi:Z\to I(x)$ be a nontrivial map, where $I(x)$ is an isogeny leaf. Non-liftability conjecture for maps into isogeny…

Let $p$ be a prime number, let $F$ be a field containing a primitive $p^2$-th root of unity, and let $n\geq 1$ be an integer. Let $\Gamma_F$ denote the absolute Galois group of $F$…

Almost-diameter conjecture. For every $\epsilon>0$, and for $q$ large enough depending on $\epsilon$, the following holds.

Let $k$ be the residue field of a discrete valuation ring $R$ of characteristic zero, and let $k[[z_2]]/k[[x]]$ be a cyclic $\Gamma$-extension with Galois sub-extension…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $\phi: Z \xrightarrow{} X$ be a cyclic $\Gamma$-cover of curves over $k$, and let $\phi_1: Y \xrightarrow{} X$…

Let $p$ be a prime, let $G e \mathbb{Z}/p^nrtimesmathbb{Z}/m$ be non-abelian with $pnmid m$, and let $k[[z]]/k[[t]]$ be a local $G$-extension whose $\mathbb{Z}/p^n$-subextension has…

Optimal strong approximation conjecture. For every $\epsilon>0$, as $q\to\infty$, there exists a set $Y\subset G_q$ with

Let $G$ be a finite group with a cyclic $p$-Sylow subgroup, and let $k[[z]]/k[[t]]$ be a $G$-extension. Cyclic Sylow lifting conjecture. For every such extension, the KGB obstructi…