Relative Fundamental Groups and Rational Points

TL;DR

This paper introduces a relative rigid fundamental group for smooth proper morphisms in characteristic p, linking it to overconvergent F-isocrystals, and explores its applications to period maps and rational points.

Contribution

It defines a new relative rigid fundamental group associated to sections of morphisms in characteristic p, with a base change property and applications to period maps.

Findings

The relative rigid fundamental group has a base change property.

Period maps can be constructed using this theory.

Targets of period maps can be interpreted as varieties in certain cases.

Abstract

In this paper we define a relative rigid fundamental group, which associates to a section $p$ of a smooth and proper morphism $f:X\rightarrow S$ in characteristic $p$, a Hopf algebra in the ind-category of overconvergent $F$-isocrystals on $S$. We prove a base change property, which says that the fibres of this object are the Hopf algebras of the rigid fundamental groups of the fibres of $f$. We explain how to use this theory to define period maps as Kim does for varieties over number fields, and show in certain cases that the targets of these maps can be interpreted as varieties.