Congrats to C. Y. Chang, a number theorist working on transcendence in function field arithmetic (FFA)! He has been awarded a silver medal by the International Congress of Chinese Mathematicians. See this link for more.
He and his collaborators have determined all algebraic relations among various objects such as the Carlitz zeta values, and special values of an analog of the gamma function in FFA, among other things. Nothing like their work has been done yet for number fields.
Those interested in these things should again see my notes on log-algebraicity (linked just below) where some of these things are discussed briefly with references. I also recommend an internet search for Chang and his collaborators. It’s fairly easy to get a good sense of their work from the introductions to their papers.
On Tuesday I gave a talk on Anderson’s log-algebraicity at Ohio State University to an audience of mostly grad students and advanced undergrads in a seminar entitled What is…?. I used the extremely powerful transcendence results of M. Papanikolas and others as motivation for where log-algebraicity can take you. There are also applications of Anderson’s results to the new constructions of L. Taelman that we’ve been studying on this blog, but these are not mentioned below.
Notes for the talk are attached are linked RIGHT HERE. Any comments, corrections, hints, tips, or tricks are welcome.
Clearly, we’ve already been taking a bit of a break, and I (Rudy) will be away from the office for the next month. This means I won’t have a space to make any videos until later in the month of May. I plan to continue when I return, so check back in around May 20th.
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:
We’re planning a new video for next week. Sorry about the delay, we’re all just a bunch of swamped graduate students!
Actually, we have two videos in the works. In the first, Tim All will tell us some more about L. Taelman’s constructions – in particular his class module and the relation to Goss-Carlitz zeta values. In the second, we’ll have a guest Brad Waller tell us some about the peculiarities of p-adic measures and distributions. Eventually I’ll get my act together and post another video as well.
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!
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.
Stay tuned for the next video in our lecture series, which I plan to have up in the next week. Our next topic will be motivation for the Carlitz module via zeta values in positive characteristic.
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