16 problems

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

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 $k$ be a perfect field of positive characteristic. Let $f(z) in k(z)$ be a PCF map of degree $d geq 2$ such that $f'(z) neq 0$. Let $\mathfrak{p}_1, \dots, \mathfrak{p}_r$ be t…

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

Assume $2g-2+n>0$ and $n\geq 1$. Let $\mathcal{M}_{g,n}$ be the moduli stack of smooth $n$-pointed genus-$g$ curves, let $\pi:\mathcal{M}_{g,n+1}\to\mathcal{M}_{g,n}$ forget the la…

Assume $2g-2+n>0$ and $n\geq 1$. Let $\mathcal{M}_{g,n}$ be the moduli stack of smooth $n$-pointed genus-$g$ curves, let $\pi:\mathcal{M}_{g,n+1}\to\mathcal{M}_{g,n}$ forget the la…

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…

Let $\mathbf{k}[x,y,z]$ be a polynomial algebra and $\mathbf{k}\langle x,y,z\rangle$ the free associative algebra. A wild $z$-automorphism is an automorphism fixing $z$ that is not…

Let $\Omega$ be the domain, let $\mathscr{N}$ be a general target manifold with covering space $\mathscr{E}$, and let $\pi\colon\mathscr{E}\to\mathscr{N}$ be the covering map. For…

Let $d>2$ and $N\geq 2$. For an isometric embedding $\Phi\colon{\mathbb{RP}^{d-1}}\to{\mathbb R}^D$, let $C^a(N,d)$ be the constant defined in Theorem 1.1 by the supremum involving…

Failure-of-lifting conjecture. In this range, the space

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…

Let $k$ be a field of characteristic $p$, let $r$ be an integer prime to the relevant construction, and consider a cyclic $\mathbb Z/r$-extension $k[[z]]/k[[t]]$. Arithmetic Oort c…

Let $p$ be a prime, let $n$ be a positive integer, and let $D_{p^n}$ denote a dihedral group of order $2p^n$. A local action of $D_{p^n}$ is an action on a formal power series ring…