4 problems

Let $U$ be a smooth curve over $k$ with compactification $C$. Let $M$ be an irreducible overconvergent $F$-isocrystal on $U$ whose fiber Newton polygon is constant as $x$ varies in…

Wan's genus-stability conjecture. If $\rho$ is algebraic-geometric and ordinary, then the sequence $(g_i)$ is given by a polynomial in $p^i$ for $i\gg 0$.

Slope stability conjecture. The following hold: (1) the $q$-slopes are equidistributed in $[0,1]$ as $n\to\infty$; (2) for each fixed rational $\alpha\in[0,1]$, there is a polynomi…

Genus stability conjecture. There is a polynomial $G(x)\in\mathbb{Q}[x]$ of degree at most ${\rm \dim}(G)+1$, depending on the tower, such that for all sufficiently large $n$,