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

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

Zbl 0721.00006