On the existence of non-geometric sections of arithmetic fundamental groups

TL;DR

This paper demonstrates the existence of non-geometric sections of certain arithmetic fundamental groups of hyperbolic curves over p-adic fields, revealing new insights into their structure beyond rational points.

Contribution

It establishes the existence of non-geometric sections in the arithmetic fundamental group quotients, a novel result in the study of hyperbolic curves over p-adic fields.

Findings

Existence of non-geometric sections proven

Sections do not originate from rational points

Advances understanding of fundamental group structures

Abstract

We show the existence of group-theoretic sections of the "etale-by-geometrically abelian" quotient of the arithmetic fundamental group of hyperbolic curves over $p$-adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points.