8 problems

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with non-trivial valuation, let $M$ be a finite free differential module over $K[\![t]\!]_0$, and let…

Nakayama's local epsilon conjecture. There exists an isomorphism

Let $K$ be the coefficient field, let $m$ be a positive integer, and let $\epsilon\in(0,1)$. Let $A_K[\epsilon,1)^m$ denote the corresponding polyannulus, and let $\mathcal{E}$ be…

Let $I$ be open, let $X$ be connected and affinoid with a finite Shilov boundary, and let $\mathcal{E}$ be a regular connection of rank $n$. Suppose there are $x\in X$, a closed su…

Let $I=\{1,2\}$, let $M$ be a $\Phi$-equivariant $F$-isocrystal $(M,\varphi,\varphi_1,\varphi_2)$ in the category…

Dwork's specialization conjecture. The polygon $\mathrm{NP}_{\log,0}(D)$ is above $\mathrm{NP}_{\log,t}(D_t)$.

Dwork's conjecture.

Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$, and let $\rho:G_K\rightarrow GL(V_\rho)$ be a representation with finite local monodromy. The Artin a…