On the section conjecture over fields of finite type

Giulio Bresciani

Published 2019-11-08Version 1

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves over finitely generated extensions of $\mathbb{Q}$. This class contains a non-empty open subset of any smooth curve, and all hyperbolic ramified coverings of curves of genus at least $1$ defined over number fields. Our method also gives an independent proof of the recent result by Sa\"idi and Tyler of the fact that the birational section conjecture over number fields implies it over finitely generated extensions of $\mathbb{Q}$.