22 problems

Alternating-group monodromy conjecture. All of the $[1,1]$-origamis in Theorem have monodromy group isomorphic to $\operatorname{Alt}_{2g-1}$.

Larsen's conjecture. For all sufficiently large primes $\ell$, there exists a semisimple group scheme $\mathcal{G}_\ell$ defined over $\mathbb{Z}_\ell$, with generic fiber…

Mostow's commensurability conjecture. In each of these nine cases, $\Gamma$ is commensurable with a group obtained from the theorem describing the discrete monodromy groups of fini…

Algebraic solutions of the Painlevé VI equation are considered up to equivalence under Okamoto transformations and simple deformation. Non-abelian Schwarz list conjecture. There ar…

Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let ${\mathbf{L}}=\{{\mathcal{L}}_\lambda\}$ be an $E$-compatible system of liss…

Consider the 5-point relative pose problem with incidence projection $\pi:X\to\mathbb{C}^{20}$, where $X$ parametrizes solutions consisting of a rotation $R\in\operatorname{SO}(3)$…

Let $X$ be a smooth curve over $k$. An $F$-isocrystal on $X$ may have a monodromy group in the overconvergent setting and, when it is dagger-extendable, a corresponding dagger mono…

Persistence conjecture. This behaviour persists in all genera.

Given a compatible system $\{\rho_\ell\}_\ell$ and its algebraic monodromy groups $\mathbf{G}_\ell\subset\mathrm{GL}_{n,\mathbb{Q}_\ell}$ and images…

Let $f:X\rightarrow \mathbb P^1$ be an indecomposable covering of degree $n$, with $X$ of genus $g_X=g$, and let $\operatorname{Mon}(f)$ denote its monodromy group. For each intege…

Zarhin's microweight conjecture. The microweight phenomenon should pertain when $\operatorname{char}(F)=p>0$.

Let $f\colon V\hookrightarrow U$ be an open sub-curve, let $u\in V({\mathbb{F}}_{q^e})$ be a base point, and let ${\mathcal{F}}$ be a convergent $F$-isocrystal on $U$. Write…

Let ${\mathcal{F}}$ be a convergent $F$-isocrystal on the curve $U$, fix $u\in U({\mathbb{F}}_{q^e})$, and let $\operatorname{Gr}({\mathcal{F}}/U,u)$ be its monodromy group over…

Let $U$ be the curve, let $K$ be the specified finite totally ramified extension, and let ${\mathcal{F}}$ be a convergent $F$-isocrystal on $U$. Fix $u\in U({\mathbb{F}}_{q^e})$, a…

Let $d$ be an integer satisfying … The two conditions referred to in the preceding construction are assumed to hold for the associated surfaces and cylinder waist curves. Quaternio…

Let $f$ be the family under consideration, let $X=f^{-1}(s)$ be a fiber, let $H^n(X,\mathbb{Z})_{(1)}$ denote the relevant eigenspace, let $Q$ be its polarization, and let $\Gamma$…

Consider the 14 rank-$4$ hypergeometric local systems that could arise as the middle cohomology of families of Calabi–Yau threefolds with $h^{2,1}=1$. For these systems, let … be t…

Let a flat bundle over a complex curve have Lyapunov exponents, and let the degree refer to the degree of a holomorphic subbundle of the bundle. Let the monodromy group be the subg…

Let $H(\alpha,\beta)$ denote a hyperbolic hypergeometric monodromy group, where $\alpha$ and $\beta$ are the parameter tuples defining the corresponding hypergeometric equation. A…

Let $K$ be a number field, let $\overline{K}$ be its algebraic closure, and write $G_K:=\operatorname{Gal}(\overline{K}/K)$. Let $\mathscr{P}$ be the set of prime numbers, and let…

In the setting of stable kernel bundles of degree $0$ and rank $r$ on a smooth curve in ${\mathds{P}}^r$, assume that the Tannaka dual group of the restriction is connected. The pr…

Let $g$ range over the relevant genus parameter, and consider covers of $\mathbb P^1_z$ whose monodromy groups arise in the genus-zero setting. A composition factor is a simple fac…