Drinfeld-Stuhler modules and the Hasse principle

TL;DR

This paper develops a theory of isogeny characters for Drinfeld-Stuhler modules and uses it to identify conditions under which certain modular varieties lack rational points, providing explicit examples violating the Hasse principle.

Contribution

It introduces a new theory of canonical isogeny characters for Drinfeld-Stuhler modules and applies it to study rational points on modular varieties.

Findings

Explicit criteria for non-existence of rational points

Construction of Drinfeld-Stuhler curves violating the Hasse principle

Development of a theory analogous to that for abelian surfaces

Abstract

We develop a theory of canonical isogeny characters of Drinfeld-Stuhler modules similar to the theory of canonical isogeny characters of abelian surfaces with quaternionic multiplication. We then apply this theory to give explicit criteria for the non-existence of rational points on Drinfeld-Stuhler modular varieties over the finite extensions of $\mathbb{F}_q(T)$. This allows us to produce explicit examples of Drinfeld-Stuhler curves violating the Hasse principle.