On the cuspidalization problem for hyperbolic curves over finite fields

TL;DR

This paper investigates the group-theoretic structures of hyperbolic curves over finite fields, demonstrating how isomorphisms between fundamental groups influence cuspidalization problems and the structure of these curves.

Contribution

It establishes that Frobenius-preserving isomorphisms between fundamental groups induce isomorphisms for curves with different points removed, advancing understanding of cuspidalization in this context.

Findings

Frobenius-preserving isomorphisms induce isomorphisms between fundamental groups of related curves.

Results apply to cuspidalization problems for hyperbolic curves over finite fields.

Provides new insights into the structure of arithmetic fundamental groups.

Abstract

In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobenius-preserving isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-$l$ fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.