Anderson-Ihara theory: Gauss sums and circular units. (English) Zbl 0733.14012

Coleman, R. F.

Algebraic number theory - in honor of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values L-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 55-72 (1989).

[For the entire collection see Zbl 0721.00006.]
Let \(\Gamma (s)=\int^{\infty}_{0}u^ se^{-u}\frac{du}{u} \) be Euler’s function and \(B(s,t)=\int^{\infty}_{0}u^{s-1}(1-u)^{t- 1}du.\) These classical functions satisfy many functional equations such as: \((1)\quad B(s,t)=B(t,s),\) \((2)\quad B(r,s)B(r+s,t)=B(r,s+t)B(s,t),\) and, \((3)\quad B(s,t)=\Gamma (s)\Gamma (t)/\Gamma (s+t).\) From a “motivic” point of view, these equations reflect the “factorization” of Fermat curves. To analyze this factorization, G. Anderson introduced the “hyperadelic \(\Gamma\)-function”.
On the other hand, Y. Ihara discovered a new sort of power series related to the Galois action on Tate-modules of the Fermat curves: \(X^{\ell^ n}+Y^{\ell^ n}+Z^{\ell^ n}=0\). Anderson then showed how these power series are the “Beta”-analogs of his hyperadelic \(\Gamma\)-functions with an appropriate factorization. Moreover, there are the expected relationships with Gauss and Jacobi sums.
In the paper under review, the author presents a discussion of these “hyper”-functions and gives some of their applications to cyclotomy.

Reviewer: D.Goss (Columbus)

MSC:

14H25	Arithmetic ground fields for curves
33B15	Gamma, beta and polygamma functions
11R18	Cyclotomic extensions
14H45	Special algebraic curves and curves of low genus
14A20	Generalizations (algebraic spaces, stacks)
11R23	Iwasawa theory
11L05	Gauss and Kloosterman sums; generalizations

Keywords:

Gauss sums; Beta-function; hyperadelic \(\Gamma \) -function; Tate-modules of the Fermat curves; Jacobi sums; cyclotomy