27 problems

Extension conjecture. Every $\rho\in{\sf Fib}_k(U)$ extends to a functor in ${\sf Fib}_k(V)$.

Let $K$ be a number field and let $X$ and $X'$ be smooth hyperbolic curves defined over $K$. Denote by $\pi_1^{mot}(\bar X)$ the motivic fundamental group over $\bar K$, and let…

Let $\mathbb P^1-\{0,1,\infty\}$ be the thrice-punctured projective line over $\mathbb Q$, let $Gal(\overline{\mathbb Q}/\mathbb Q)$ be its absolute Galois group, and let…

Let $X$ be a smooth projective rigid analytic space over $\mathbb{C}_p$, and let $G$ be a reductive group defined over $\mathbb{C}_p$. A $G(\mathbb{C}_p)$-representation of…

Let $X$ and $Y$ be smooth, geometrically connected varieties over … mathbb{C}$. Here$X($\mathbb{C})^{\sim}$ and $Y($mathbb{C})^{sim}$denote their schematic homotopy types, and$m…

Let $V/k$ be a smooth, geometrically connected, hyperbolic curve over a number field $k$, with smooth completion $V\subset X$ and boundary $Y=X\setminus V$. For each $k$-rational c…

Kummer-faithfulness conjecture. Any finite extension of $\overline{K}(\sigma)$ is Kummer-faithful for almost all $\sigma\in G_K^e$.

Let $U_i$ be the genus-zero hyperbolic curves and let $\Pi_{U_i}^{(2,\ell)}$ be the groups used in the source for a prime $\ell$. Let $(\overline{\Pi}_{U_i}^{2,\ell})^{[1]}$ denote…

Let $k$ be a field with absolute Galois group $G_k$, let $U_1,U_2$ be the curves under consideration, and let $m,n$ be non-negative integers. For each $r$, write…

Let $k$ be a finitely generated field of characteristic $p>0$ with absolute Galois group $G_k$, let $m$ be a non-negative integer, and let $U_i$ be hyperbolic curves over $k$ for…

Let $k$ be a finitely generated field of characteristic zero with absolute Galois group $G_k$, let $m$ be a non-negative integer, and let $U_i$ be hyperbolic curves over $k$ of typ…

Let $k$ be a field of characteristic $p\geq 0$, finitely generated over the prime field, with absolute Galois group $G_k$. Let $U$ be an anabelian variety over $k$ with a geometric…

Grothendieck's conjecture. The variety $U$ may be reconstructed group-theoretically from $\pi^{\acute{e}\mathrm{t}}_1(U,\ast)$ and this projection. The conjecture for hyperbolic cu…

Let $K$ and $L$ be finite extensions of $\mathbb{Q}_p$, and let $X/L$ and $Y/K$ be hyperbolic curves. Write…

Let $C$ and $D$ be smooth projective curves of equal genus at least $2$ over a finite field ${\mathbb F}$. Embed $C$ into its Jacobian and define its Hilbert class field cover…

Anabelian characterization conjecture. The subgroup $\Delta_X$ is the unique closed normal subgroup $N$ of $\Pi_X$ satisfying: (i) $\Pi_X/N\simeq\mathbb{Z}_p$; and (ii) for every o…

Let $K_i$ be number fields for $i=1,2$, let ${\mathcal C}_i$ be nontrivial full classes of finite groups, and let … be an isomorphism. Here $\Sigma({\mathcal C}_i)$ denotes the ass…

Let $M_{g,n}^{\rm ord}$ be the ordinary moduli space, let $q_{\rm gen}$ be its generic point, and let $\pi_A^{\rm t}(q)$ denote the set of finite quotients associated with the tame…

Let $M_{g,n}$ be the ordinary moduli space of smooth pointed stable curves and let $M_{g,n,\mathbb{F}_p}$ be the corresponding characteristic-$p$ moduli space. For arbitrary points…

Let $M_{g,n}^{\rm ord}$ be the ordinary moduli space of smooth pointed stable curves, and let $M_{g,n,\mathbb{F}_p}$ be its characteristic-$p$ counterpart. For arbitrary points…

Let $p>0$ be the characteristic. Retain the Frobenius-quotiented moduli space $\overline{\mathfrak{M}}_{g,n}$ of pointed stable curves of type $(g,n)$ and let…

Let $\Bbbk$ be a sub-$p$-adic field with absolute Galois group $G_\Bbbk$. Let ${\mathcal M}^{\lambda}_\Bbbk$ and ${\mathcal M}^{\mu}_\Bbbk$ be moduli stacks of curves with congruen…

Let $k$ be a finitely generated field over $\operatorname{\mathbb{Q}}$, and let $X$ be a smooth, projective, geometrically connected curve over $k$. Write…

Let $k$ be finitely generated over $\mathbb{Q}$. Let $T/k$ be a smooth variety, and let $X/k$ be a smooth, proper, geometrically connected hyperbolic curve. Grothendieck's Hom conj…

Let $k$ be a number field or a finite extension of $\mathbb Q_p$, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$, with function field…