Sections, Homotopy Rational Points and Reductions of Curves

TL;DR

This paper investigates the relationship between unramified sections of the fundamental group of smooth projective curves over p-adic fields and their special fibers, revealing how specialized sections induce unique homotopy rational points under mild conditions.

Contribution

It establishes a connection between sections of the fundamental group and homotopy rational points on special fibers, providing new insights into their compatibility and applications in cohomology.

Findings

Specialized sections induce unique homotopy rational points under mild assumptions.

Compatibility between generic and special fiber sections in cohomological contexts.

Applications to the $\, ext{l}$-adic cycle class of a section.

Abstract

We study unramified sections of the fundamental group sequence of smooth projective curves of genus $\geq 2$ over $p$-adic fields together with an integral model. We are particularly interested in the induced specialized sections of the special fibre and how they relate to homotopy rational points over the residue field. Under mild assumptions, such a specialized section induces a unique homotopy rational point of the special fibre that is compatible with the original section of the generic fibre in cohomological settings. We give two applications of such specialized homotopy rational points around the $\ell$-adic cycle class of a section.