[NTNU Number Theory Seminar] 【07月27日】于靖 / The special Gamma Values —From n! to Crystalline. Periods

ABSTRACT

Euler interpolated n! to the classical Gamma function with Γ(n+1) :=n!. Since Γ(1/2) = √π is transcendental, one wanders about the Gamma values at other proper fractions. In 1976 G. V. Chudnovsky succeeded in proving Γ(1/3), Γ(1/4) are also transcendental and algebraically independent from π. In his work the periods of specific elliptic curves play a vital role. One actually conjectures that the Gamma values at all proper fractions are transcendental and Lang-Rohrlich even speculates that all algebraic relations among these transcendental special values come from well-known functional equations of Euler’s Gamma function. 

Y. Morita in 1975 looked at the n! from p-adic arithmetic, p any fixed prime number. He introduced p-adic Gamma function Γp defined on Zp with values in the algebraic closure of Qp satisfying Γp(n + 1) = −nΓp(n), if p ̸= n. This case Γp(1/2) = √±1 is always algebraic.
In 1979, Gross-Koblitz discovered that if n ≡ 1 (mod p), all proper fractions with denominator n turns out to be algebraic. This leads to conjecture that the values of the p-adic Gamma at the other proper fractions should be transcendental, as they are related to the crystalline
periods of certain abelian varieties by Ogus 1989.

We are interested in factorials and Gamma functions for function fields in the positive characteristic worlds, after Carlitz, Goss, Thakur, Anderson, Brownawell, and Papanikolas. Particularly I will report on recent progress of special v-adic Gamma values for finite prime v, and the ongoing work by Chieh-Yu Chang, Fu-Tsun Wei and me.