19 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 $K[\![t]\!]_0$ admittin…

Dwork's conjecture. The horizontal sections satisfy

Let $k$ be the field of constants, let $Y$ be a smooth algebraic variety over $k$, let $Z$ be a divisor in $Y$, and let $({\mathcal M},\nabla) \in {\bf MIC}((Y \setminus Z)/k)$. Fo…

A $p$-adic differential equation is a finite-rank free module $M$ over the Robba ring $\mathcal{R}_K$, equipped with a connection $\partial_M:M\rightarrow M$. It has a Frobenius st…

Crew's conjecture. Every $p$-adic differential equation on an annulus with Frobenius structure has quasi-unipotent monodromy.

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

Nakayama's local epsilon conjecture. There exists a unique 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…

A differential equation is considered together with its solution space and the associated Frobenius action. A Frobenius structure means a structure expressing compatibility between…

Let $K[[x]]_0$ denote the ring of analytic functions on the unit open disc with log-growth at most $0$, and for $\lambda\geq0$ let $K[[x]]_\lambda$ be the corresponding space of fu…

Chiarellotto–Tsuzuki conjecture. All slopes of the log-growth filtration $V(M)^\bullet$ are rational, and

Let $\mathbb{K}$ be a number field, and let $L\in\mathbb{K}[z,\frac{{\rm d}}{{\rm d}z}]$ be a G-operator, meaning that its differential equation has a $G$-function solution for whi…

Chiarellotto–Tsuzuki's conjecture. The breaks of the filtration $\operatorname{Sol}_{\bullet}(D)$ are rational, and

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

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$, let $G_K$ be its absolute Galois group, and consider $p$-adic representations of $G_K$. A representa…

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…