Local diophantine properties of modular curves of $\cal{D}$-elliptic sheaves

TL;DR

This paper investigates the local properties of modular curves associated with $\\cal{D}$-elliptic sheaves, focusing on rational points, special fibers, and applications to arithmetic groups and Hasse principle violations.

Contribution

It provides new insights into the structure and rational points of these modular curves, including explicit equations and applications to arithmetic group presentations.

Findings

Criteria for the existence of rational points on the curves

Descriptions of the special fibers of the modular curves

Examples of curves violating the Hasse principle

Abstract

We study the existence of rational points on modular curves of $\cal{D}$-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of $\cal{D}$-elliptic sheaves, and constructing curves violating the Hasse principle.