Mass formula of division algebras over global function fields

TL;DR

This paper provides two proofs of the mass formula for definite central division algebras over global function fields, connecting Tamagawa measures and zeta functions to establish the formula.

Contribution

It introduces two novel proofs of the mass formula, one via Tamagawa measures and another through analytic methods involving zeta functions.

Findings

Two rigorous proofs of the mass formula are presented.

The relationship between mass and zeta function values is established.

The proofs deepen understanding of division algebras over function fields.

Abstract

In this paper we give two proofs of the mass formula for definite central division algebras over global function fields, due to Denert and Van Geel. The first proof is based on a calculation of Tamagawa measures. The second proof is based on analytic methods, in which we establish the relationship directly between the mass and the value of the associated zeta function at zero.