The Section Conjecture for Graphs and Conical Curves

TL;DR

This paper formulates and proves a combinatorial version of the section conjecture for finite groups acting on graphs, and applies it to rational points on certain algebraic curves, showing finite descent is the only obstruction to the Hasse principle.

Contribution

It introduces a combinatorial approach to the section conjecture for graphs and applies it to rational points on specific algebraic curves, linking group actions and arithmetic geometry.

Findings

Proves a combinatorial version of the section conjecture for finite groups on graphs.

Shows finite descent is the sole obstruction to the Hasse principle for certain singular curves.

Establishes a connection between graph symmetries and rational points on algebraic curves.

Abstract

In this paper we formulate and prove a combinatorial version of the section conjecture for finite groups acting on finite graphs. We apply this result to the study of rational points and show that finite descent is the only obstruction to the Hasse principle for mildly singular curves whose components are all geometrically rational.