Further examples of non-geometric sections of arithmetic fundamental groups

TL;DR

This paper demonstrates the existence of non-geometric sections in certain arithmetic fundamental groups of hyperbolic curves over p-adic fields, revealing new group-theoretic phenomena beyond rational point origins.

Contribution

It provides explicit examples of non-geometric sections in geometrically pro-nilpotent by abelian arithmetic fundamental groups, including the geometrically metabelian case.

Findings

Existence of non-geometric sections in specific fundamental groups

Examples include geometrically metabelian fundamental groups

Results extend understanding of fundamental group sections over p-adic fields

Abstract

We show the existence of group-theoretic sections of certain geometrically pro-nilpotent by abelian arithmetic fundamental groups of hyperbolic curves over p-adic local fields which are non-geometric, i.e., which do not arise from rational points. Among these quotients is the geometrically metabelian arithmetic fundamental group.