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The curve $X_1(Np^n)$ is connected, but the ordinary locus in this curve is not: if you remove the residue discs of the supersingular points, what's left "falls apart" into a disjoint union of several …

There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications? For the first question, it' …

This is an algebraic question disguised as a rigid-analytic one: what you're asking for is a description of the $\mathbf{Z}_p$-scheme $X^{\star} - X$, the boundary of the arithmetic minimal compactifi …

Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball. Consider the sets $U := \{ |T_1| < 1\}$ and $V := …

Let $K$ be a $p$-adic field with residue field $k$, and $X$ a variety over $k$. The rigid cohomology of $X$ over $k$ can be described very neatly using Grosse--Kloenne's notion of dagger spaces: embed …

If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ther …