On the section conjecture and Brauer-Severi varieties

TL;DR

This paper extends the understanding of the section conjecture by linking it to Brauer-Severi varieties and Weil restrictions, showing that certain rational maps imply the conjecture holds for the curve.

Contribution

It generalizes previous results by proving the section conjecture for curves over number fields with Weil restrictions mapping to Brauer-Severi varieties, involving corestriction conditions.

Findings

Section conjecture holds under Weil restriction conditions.

Rational maps to Brauer-Severi varieties imply the conjecture.

Corestriction non-triviality ensures the conjecture for the curve.

Abstract

J. Stix proved that a curve of positive genus over $\mathbb{Q}$ which maps to a non-trivial Brauer-Severi variety satisfies the section conjecture. We prove that, if $X$ is a curve of positive genus over a number field $k$ and the Weil restriction $R_{k/\mathbb{Q}}X$ admits a rational map to a non-trivial Brauer-Severi variety, then $X$ satisfies the section conjecture. As a consequence, if $X$ maps to a Brauer-Severi variety $P$ such that the corestriction $\operatorname{cor}_{k/\mathbb{Q}}([P])\in\operatorname{Br}(\mathbb{Q})$ is non-trivial, then $X$ satisfies the section conjecture.