[NTNU Number Theory Seminar] 【11月24日】于靖 / Quasi-periods, and the Chowla-Selberg Phenomenon

I will tell a story developed in the last three decades. Chronologically this is what happens. 1987, starting from the Carlitz module with complex multiplications, Deligne, Yu, Anderson, and Gekeler discovered the quasi-periods theory for Drinfeld modules. Yu also showed the transcendence of the non-zero quasi-periods for all the Drinfeld modules . In the meantime, Thakur developed the gamma functions for the rational function fields in finite characteristic, arithmetic gamma (following Goss) as well as geometric gamma. In the 1991 Annals paper, Thakur worked out a formula for the Carlitz module with CM, connecting its periods to special arithmetic gamma values. Developments in 1990’s by Yu, Thakur, Sinha, Brownawell, Papanikolas proved all “special” gamma values in the function field world are transcendental. Thakur was led to conjecture/recipes asking for formulas
expressing abelian CM periods in terms of appropriate special gamma values, i.e. the Chowla-Selberg phenomenon.

2004, Anderson, Brownawell, and Papanikolas determined all the algebraic relations among special geometric gamma values (the Lang-Rohrlich conjecture). 2010, Chang, Papanikolas, Thakur, and Yu determined all the algebraic relations among special arithmetic gamma values. They also exhibit a Chowla-Selberg phenomenon for specific basis of quasi-periods of Carlitz modules with CM, and verified that the transcendence degree of the field generated by all quasi-periods equals to the rank, with canonical transcendence( also linear) basis given by explicit special arithmetic gamma values.

2011-12, Chang and Papanikolas prove that the analogue of the Legendre’s relation gives rise to the only algebraic relations among quasi-periods apart from the possible complex multiplications. For all CM Drinfeld modules, fields generated by quasi-periods always have transcendence degree equal to the rank. 2022, Brownawell, Chang, Papanikolas, and Wei developed a complete period symbol theory in the function field setting. They prove in particular the analogue of Shimura’s algebraic independence conjecture. Finally in 2022, Wei verifies a strong Chowla-Selberg phenomenon for all Abelian CM Drinfeld modules, giving an explicit linear as well as transcendence basis of quasi-periods for any such Drinfeld module in terms of “appropriate” product of special gamma values. Appropriateness here means that the arithmetic invariants of the CM field in question are contained in the recipe for taking product.

In this talk we shall look closely at the fascinating connections between periods and special Gamma values. These connections are explained by Arithmetic Geometry (algebraic number theory together with algebraic geometry). Particularly we focus on the route from explicit class field theory of Kronecker-Weber to the Complex Multiplication period symbol of Shimura.