Speaker：于靖 院士
Job title：台灣大學名譽教授、清華大學榮譽講座教授、台灣師範大學講座教授
Title : Periods, Gamma values, and Algebraic Relations
Abstract：
This is a survey talk, leading to recent developments and open problems.
The story started four centuries ago, with Euler, Gauss, Legendre, … The simplest period is $2\pi\sqrt{-1}$, as every mathematics student knows. We view periods as transcendental invariants attached to arithmetical objects via “analytic” means, e.g. arc length of the unit circle. We are interested in understanding these natural transcendental quantities, or numbers. Facing the problem of finding all algebraic relations among the transcendental invariants. If several numbers arise naturally, and no one knows any algebraic relations in between these numbers, we wish to prove that they are actually algebraically independent. In the sense that verifying these, say N, numbers are not roots of any non-zero polynomial of N variables with coefficients from $\mathbf Z$, when you substitute the N numbers for the N variables into polynomials, you never get zero.
Periods of arithmetical objects should be algebraically independent in general, unless there are natural precious algebraic relations coming from the “geometry” behind the scene. This is the 20 century philosophy leading to conjectures of Grothendieck, Deligne, Shimura, Lang-Rohrlich…. In the positive characteristic worlds, the global function fields over finite fields play roles as number fields, natural arithmetic objects also abound. The algebraic dependence/independence phenomena in Arithmetic Geometry here are a little more transparent. One can manage to prove more. I will give a very brief introduction in the end of this talk on the algebraic independence of all coordinates of a non-zero period vector of a Hilbert-Blumenthal-Drinfeld module with complex multiplication, recent work of Brownawell-Chang-Papanikolas-Wei (extending a theorem of mine 30 years ago) .
Time: 1:30 p.m., March 25(Fri.), 2022
Place: Room 210, Department of Mathematics, NTNU