6 problems

Let $C$ be a smooth projective curve, let $S\subset C$ be a finite subset, and fix an integer $q\geq 2$. Let $\mathbb T$ denote the parameterizing scheme for non-isotrivial familie…

Let $C'$ be a semistable curve over $F$, and let $C_L$ be the model associated with a finite nonramified extension $L$ of $F(C')$. Geometric Shafarevich conjecture. There is a numb…

Let $K(C)$ be a field as above, and let $\rho$ be an ideal of the field $F^{\mathrm{nr}}$. Strong arithmetic Shafarevich conjecture. There exists a finite extension $F$ of $K$ and…

Pointed Shafarevich conjecture. The set of morphisms $f:C\to U$ satisfying $f(c)=u$ is finite. This predicts finiteness of families over a curve after fixing one fibre over one mar…

Let $K=K^lK_l$ be a neat compact open subgroup of $G(\mathbb A_f)$, let $x=[s,\overline{1}]$ be a point of $Sh_K(G,X)(L)$, and fix a lift $\widetilde{x}=[s,1]$ in $Sh_{K^l}(G,X)$.…

Higher-dimensional Shafarevich conjecture. (B) Families of canonically polarized manifolds over $U$ with Hilbert polynomial $h$ fall into finitely many deformation equivalence clas…