On the section conjecture over fields of finite type

TL;DR

This paper extends the validity of the section conjecture from number fields to a broad class of curves over finitely generated extensions of , assuming certain conjectures, thus broadening its scope significantly.

Contribution

It proves the section conjecture for many curves over finitely generated fields, assuming it holds over number fields, and further extends results under the weak Bombieri-Lang conjecture.

Findings

Section conjecture holds for hyperelliptic and low-genus curves over finitely generated fields.

Extension of the conjecture's validity to a broad class of curves.

Conditional proof of the conjecture for all hyperbolic curves assuming the weak Bombieri-Lang conjecture.

Abstract

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $\mathbb{Q}$. This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus $\le 2$, and a non-empty open subset of any curve. If we furthermore assume the weak Bombieri-Lang conjecture, we prove that the section conjecture holds for every hyperbolic curve over every finitely generated extension of $\mathbb{Q}$.