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We are interested in special values, e.g. the family of Gamma values taken at proper fractions Q – Z. These values are supposed to be transcendental numbers with “standard” algebraic relations among them (the Lang-Rohrlich Conjecture).

On the other hand, we have the family of periods coming from various structure in mathematics, e.g. 2 pi i. Both these two families of special values are not “well-understood”. But we do know certain connections between them, started with the fact that Gamma value at 1/2 equals to the square root of pi. We know a little bit more about the transcendence of periods, after Lindemann,  Siegel, … Further connections of special Gamma values with “elliptic” periods enable us to confirm the transcendence of Gamma values at 1/3, 1/4, and 1/6 (Chudnovsky 1976).

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.