3 problems

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 $0<\alpha<1$, let $\epsilon>0$, and let $k=3$ or $4$. For a positive integer $N$, write $[N]=\{1,2,\dots,N\}$. A length-$k$ arithmetic progression in $[N]$ has a common differe…