# Valentijn Karemaker: Comparing obstructions to local-global principles for rational points over semiglobal fields

**Time: **
Wed 2019-12-04 11.00 - 12.00

**Location: **
KTH, F11

**Participating: **
Valentijn Karemaker, Universiteit Utrecht and Stockholms Universitet

### Abstract

Let \(K\) be a complete discretely valued field, let \(F\) be the function field of a curve over \(K\), and let \(Z\) be a variety over \(F\). When the existence of rational points on \(Z\) over a set of local field extensions of \(F\) implies the existence of rational points on \(Z\) over \(F\), we say a local-global principle holds for \(Z\).

In this talk, we will compare local-global principles, and obstructions to such principles, for two choices of local field extensions of \(F\). On the one hand we consider completions \(F_v\) at valuations of \(F\), and on the other hand we consider fields \(F_P\) which are the fraction fields of completed local rings at points on the special fibre of a regular model of \(F\).

We show that if a local-global principle with respect to valuations holds, then so does a local-global principle with respect to points, for all models of \(F\). Conversely, we prove that there exists a suitable model of \(F\) such that if a local-global principle with respect to points holds for this model, then so does a local-global principle with respect to valuations.

This is joint work with David Harbater, Julia Hartmann, and Florian Pop