Abstract

The aim of this series of lectures is to give introduction, accessible to graduate students, to the topic of modular varieties arising from
central division algebras over function fields. The idea of these varieties was proposed by Ulrich Stuhler as a natural generalization of
Drinfeld modular varieties. These varieties were then used by Laumon, Rapoport, and Stuhler to prove the local Langlands correspondence in
positive characteristic. We will mostly concentrate on the case of Drinfeld-Stuhler modular curves, which are the analogues of Shimura
curves arising from indefinite quaternion algebras over the rationals.

Lecture 1 and 2: We will introduce Drinfeld-Stuhler modular curves analytically as Mumford curves and discuss their basic invariants and
their theory of modular forms. We then discuss the algebraic objects parametrized by these curves.

Lecture 3 and 4: We will discuss some arithmetic properties of Drinfeld-Stuhler modular curves, such as the existence of rational
points over local and global fields. One of the applications of this is the construction of curves over global fields violating the Hasse
principle.

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