68 problems

Grothendieck conjecture. The geometry of anabelian schemes over fields finitely generated over $\mathbb{Q}$ is completely determined by their étale fundamental groups; in particula…

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

Grothendieck's section conjecture. Every section $s_X\in\mathcal{S}(\pi_1(X)/\pi_1(k))$ arises from a point $x\in X(k)$, and

Grothendieck's section conjecture. The section map

Grothendieck's section conjecture. The natural map $C(k)\to\Pi_{C/k}(k)$ is bijective.

Let $E$ and $E'$ be $p$-adic fields, and let $Y/E'$ and $X/E$ be anabelomorphic, geometrically connected, smooth hyperbolic curves. Here, anabelomorphic means that their associated…

Homeomorphism conjecture. The map $\pi_{g,n}^{\rm adm}$ is a homeomorphism.

Let $C$ be a proper smooth curve of genus $g$ over a field $k$ finitely generated over ${\mathbb Q}$. Choose a geometric point $\mathfrak b x$ of $C$, set…

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

Section conjecture reformulated. The map

Grothendieck's section conjecture. If $k$ is of finite type over $\mathbb{Q}$, then: (SC1) the map

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

Let $F$ be a number field and let $X$ be a smooth, compact, hyperbolic curve over $F$. Choose an algebraic closure $\bar F$, a geometric base point $x$ of $X$, and write…

Let $K$ be a local $p$-adically closed field, meaning a finite extension of $\mathbf{Q}_p$, and let $X$ be a smooth complete curve of genus greater than $1$ over $K$. Let…

General Section Conjecture. Both inclusions

Fenchel's conjecture. Every orbicurve admits a topological finite cover by a variety.

Let $K$ be a field finitely generated over $\mathbb{Q}$, let $X$ be a smooth variety over $K$, and let $Y$ be a hyperbolic polycurve over $K$. Write…

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…

Grothendieck's section conjecture. Every section of this sequence is either geometric or cuspidal.

Let $C$ be a nonsingular geometrically connected hyperbolic curve over a number field $F$, with function field $F(C)$. Let $\overline{F}^{\mathrm{sep}}$ be a separable closure of…

Let $k$ be a finitely generated field of characteristic $0$. Let $\operatorname{Sch}^{\mathrm{ft}}_k$ be the category of finite type $k$-schemes, and let…

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…