The volume of a locally symmetric space is essentially a product of special values of zeta functions. Its connection with automorphic forms featured in early work of Piatetski-Shapiro. There is a natural function field analog of the volume given by counting principal bundles on an algebraic curve over a finite field. 

 

Continuing in the function field setting, but going further in the arithmetic direction, we consider the volume of the moduli space of Shtukas (function field analog of Shimura varieties). I will explain how to make sense of the apparently divergent volume, and express the result in terms of higher derivatives of a curious variant of zeta functions. This is joint work with Tony Feng and Wei Zhang.