Rational points and Brauer--Manin obstruction on Shimura varieties of level one classifying abelian varieties with quaternionic multiplication

TL;DR

This paper investigates the scarcity of rational points on certain Shimura varieties that classify abelian varieties with quaternionic multiplication, highlighting the role of the Brauer--Manin obstruction.

Contribution

It demonstrates that Shimura varieties of level one parametrizing QM-abelian varieties rarely have rational points, advancing understanding of rational points on these varieties.

Findings

Rational points on level one Shimura varieties are rare.

The Brauer--Manin obstruction explains the scarcity of rational points.

Provides new insights into the arithmetic of QM-abelian varieties.

Abstract

We show that the Shimura varieties of level one parametrizing QM-abelian varieties have rarely rational points.