On the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups

TL;DR

This paper surveys Harder's reduction theory, connecting the geometry of global function fields, the Riemann-Roch theorem, and finiteness properties of S-arithmetic groups, highlighting key conjectures and recent progress.

Contribution

It provides a comprehensive overview of Harder's reduction theory and introduces a filtration useful for studying finiteness properties of S-arithmetic groups over global function fields.

Findings

Description of a filtration derived from Harder's reduction theory

Partial verification of the rank conjecture for S-arithmetic groups

Proposal of a general conjecture on isoperimetric properties of S-arithmetic groups

Abstract

Harder's reduction theory provides filtrations of euclidean buildings that allow one to deduce cohomological and homological properties of S-arithmetic groups over global function fields. In this survey I will sketch the main points of Harder's reduction theory starting from Weil's geometry of numbers and the Riemann-Roch theorem, describe a filtration that is particularly useful for deriving finiteness properties of S-arithmetic groups, and state the rank conjecture and its partial verifications that do not restrict the cardinality of the underlying field of constants. As a motivation for further research I also state a much more general conjecture on isoperimetric properties of S-arithmetic groups over global fields (number fields or function fields).