The Absolute Anabelian Geometry of Virtual Curves of Arbitrary Genus

TL;DR

This paper extends the study of pointed virtual curves and their fundamental groups from genus-zero to arbitrary genus, introducing new group-theoretic and categorical concepts.

Contribution

It generalizes previous anabelian results to higher genus curves and introduces the notions of CAVC-type inclusion and virtual decuspidaloid.

Findings

Established a criterion for the geometricity of virtual curves.

Provided group-theoretic conditions for sections to arise from rational points.

Extended fundamental group analysis to arbitrary genus curves.

Abstract

The objective of this paper is to further study the anabelian object referred to as \emph{pointed virtual curves}. Building upon previous work that investigated these fundamental-group-theoretic pullbacks of Galois sections in the genus-zero situation, we extend the central anabelian results to curves of arbitrary genus. To facilitate this generalization, we introduce the group-theoretic notion of an inclusion of CAVC-type and the categorical-theoretic notion of a virtual decuspidaloid. Furthermore, we establish a criterion regarding the "geometricity" of certain virtual curves, providing group-theoretic conditions under which a section of an arithmetic fundamental group arises from a rational point.