Lecture 4: Taelman’s Class Module and Connection to Goss’ Zeta

Hi Everyone,

Sorry about the delay! Here’s our latest episode in the exploration of L. Taelman’s class and unit modules for the Carlitz module.

In this video Tim gives a nice refresher on the residue at one of the Dedekind zeta function of a number field followed by the Mazur-Wiles theorem in motivation for connections of classical zeta values with class groups. He then recalls Lenny’s class module and mentions some of the connections it has with Goss zeta values.

Please be warned that there are a couple typos in this video, but Tim has graciously typed up some follow up notes which can be found at:

http://www.math.osu.edu/~perkins.298/classmoduleupdated.pdf

Enjoy!

Third Lecture: A Dirichlet Unit Theorem for the Carlitz Module

In this video Tim tells us about a (fairly) new (and long desired!) construction of L. Taelman of unit and class modules associated to a given Drinfeld module. In this lecture we focus on the Carlitz module and give all of the relevant definitions with (hopefully) beneficial motivational explanations. Tim begins by recalling the classical Dirichlet unit theorem for number fields in such a way as to make a tighter analogy when it comes to Taelman’s machine. Enjoy!

References for this video can be found on L. Taelman’s homepage, http://www.math.leidenuniv.nl/~lenny/. Of particular interest for us are the papers A Dirichlet unit theorem for Drinfeld modules and Arithmetic of characteristic p special L-values. Please check them out!

Second Lecture on the Carlitz Module

Welcome back! Here is the follow-up lecture to Tim’s introduction of the Carlitz module, this time in crisp, clear high-definition. In this second video I attempt to guide the viewer to the discovery of the Carlitz module map by following Euler’s classical calculation of the Riemann zeta values at the even integers. I would love any feedback you might have. Please let us know what you think so far by leaving a comment below. Enjoy.

First Lecture in Our Series on the Carlitz Module

We are beginning a series of lectures on the Carlitz module functor in the setting of function field arithmetic. Here Tim All gives an introduction to the Carlitz module, comparing it with the classical multiplicative group functor arising in the study of number fields. This lecture follows Rosen’s exposition on the subject, but there are many good sources that I hope to mention in later posts.

Tim has provided the beautiful slide below that should be downloaded and used to follow along with his lecture. Tim’s Slides: Lecture One