Etale fundamental groups of affinoid $p$-adic curves
Mohamed Saidi

TL;DR
This paper investigates the structure of the geometric étale fundamental group of smooth $p$-adic affinoid curves, revealing it as a semi-direct factor of a profinite free group and describing its maximal pro-$p$ and prime-to-$p$ quotients.
Contribution
It establishes the semi-direct factor relationship and characterizes the maximal pro-$p$ and prime-to-$p$ quotients of the fundamental group, providing new structural insights.
Findings
The geometric étale fundamental group is a semi-direct factor of a profinite free group.
Maximal pro-$p$ quotient is pro-$p$ free of infinite rank.
Maximal prime-to-$p$ quotient is free of finite computable rank.
Abstract
We prove that the geometric etale fundamental group of a (geometrically connected) rigid smooth -adic affinoid curve is a semi-direct factor of a certain profinite free group. We also prove that the maximal pro- (resp. maximal prime-to-) quotient of this geometric \'etale fundamental group is pro- free of infinite rank (resp. (pro-)prime-to- free of finite computable rank).
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · advanced mathematical theories
