arXiv subjects
Publications and source records attributed to Emmanuel Lepage.
Given two hyperbolic curves over p-adic local fields, the absolute anabelian conjecture claims that any isomorphism between their étale fundamental group comes from an isomorphism of schemes. This conjecture was proven by S. Mochizuki for curves of quasi-Belyi type. The goal of this paper is to extend the result to a wider family of curves, that includes for example hyperbolic Mumford curves.
Given a Mumford curve $X$ over $\bar{\mathbb Q_p}$, we show that for every semistable model $\mathcal X$ of $X$ and every closed point $x$ of this semistable model, there exists a finite étale cover $Y$ of $X$ such that every semistable model of $Y$ has a vertical component above $X$. We then give applications of this to the tempered fundamental group. In particular, we prove that two punctured Tate curves $\bar{\mathbb Q_p}$ with isomorphic tempered fundamental groups are isomorphic over $\mathbb Q_p$.
The tempered fundamental group of a p-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite etale covering of the space. Here we construct cospecialization homomorphisms between (p') versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction. In particular, we will have to study invariance of the geometric log fundamental group of saturated log smooth log schemes over a log point by change of log point.
The tempered fundamental group of a p-adic analytic variety classifies étale coverings that become topological coverings (for the Berkovich topology) after finite étale base change. I study what can be recovered from the tempered fundamental group. In particular, I prove that, for a Mumford curve, one can recover the metric of the graph of the stable reduction from the tempered fundamental group. I also study how the tempered fundamental group behaves along the fibers of a fibration.
This paper constructs cospecialization homomorphisms between the (p') versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction (the tempered fundamental group is a sort of analog of the topological fundamental group of complex algebraic varieties in the p-adic world). We studied the question for families of curves in another paper. To construct them, we will start by describing the pro-(p') tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tempered fundamental groups for log polystable varieties.
This paper is an attempt to give some general results on the tempered fundamental group of a $p$-adic smooth algebraic varieties (which is a sort of analog of the topologic fundamental group of complex algebraic varieties in the p-adic world). The main result asserts that one can recover the metric structure of the graph of the stable model of a Mumford curve from the tempered fundamental group of this Mumford curve. We will also prove birational invariance, invariance by algebraically closed extensions and a Kuenneth formula for the tempered fundamental group. We will describe the tempered fundamental group of an abelian variety and link the tempered fundamental group of a curve to the tempered fundamental group of its Jacobian variety.