Student Arithmetic Seminar

There is an old saying that it is easier to work on function fields than on number fields. A very hot research direction in number theory is to translate hard problems over number fields to function fields, and the problems generally become easier. It is then worthwhile to understand some basics of the number field - function field analogy, so as to have a glimpse on this growing area of research.

Classically, the easiest function field analogy of the Riemann zeta function is the Hasse-Weil zeta function, a complex meromorphic function encoding number of points of a curve over finite fields. In recent years, another type of analogy arises, which investigates objects called Carlitz modules and Drinfeld modules, as well as L-functions defined on them.

In this talk, we will discuss some simple analogies between number fields and function fields. We will first review basic interesting properties of Riemann zeta function. Then We will see what analogies we can say on function field, for both the Hasse-Weil zeta function and the Carlitz zeta function. This talk does not assume knowledge in number theory or algebraic geometry, though we may use some complex analysis. Speaker(s): Angus Chung (University of Michigan)