The cuspidalisation of sections of arithmetic fundamental groups II

TL;DR

This paper explores the cuspidalisation process of sections of arithmetic fundamental groups of hyperbolic curves, providing criteria for sections to originate from rational points and identifying special classes of such sections over p-adic fields.

Contribution

It introduces new conditions for sections to come from rational points and identifies classes of sections orthogonal to Pic over p-adic curves.

Findings

Two necessary and sufficient conditions for sections to arise from rational points.

Identification of sections orthogonal to Pic that satisfy one of the conditions.

Advancement in understanding the structure of arithmetic fundamental groups of hyperbolic curves.

Abstract

In this paper we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally i-th and 2/p-th step prosolvable arithmetic fundamental groups. As a consequence we exhibit two, necessary and sufficient, conditions for sections of arithmetic fundamental groups of hyperbolic curves over p-adic local fields to arise from rational points. We also exhibit a class of sections of arithmetic fundamental groups of p-adic curves which are orthogonal to Pic, and which satisfy (unconditionally) one of the above conditions.