On Ihara's power series
type: Item
instance of: scholarly article
title: On Ihara's power series
zbMATH Open document ID: 0838.11079
author: Hiroo Miki
published in: Journal of Number Theory
publication date: 1996-06-03T00:00:00Z
review text: \textit{Y. Ihara} [Ann. Math., II. Ser. 123, 43-106 (1986; Zbl 0595.12003)] found a power series \(F_\sigma (u,v) \in \mathbb{Z}_\ell u,v\) which interpolates Jacobi sums, and further determined the power series explicitly under certain conditions. He further conjectured its validity in generality. Subsequently, Ihara's conjecture has been proved by several mathematicians including Ihara, Anderson, Coleman, Miki and Ihara et al. In this paper, a new, simpler proof of Ihara's conjecture is presented, using the congruence on Gauss sums. In fact, the author introduces the Gauss sum power series \(G_{\ell^n} (T)\) \((\ell\) a prime) generalizing the earlier work on Gauss sums [the author, Adv. Stud. Pure Math. 12, 87-118 (1987; Zbl 0649.12002)]. In particular, he determines explicitly the coefficients of the Gauss sum power series modulo \(\ell^n \). The idea of proof is exactly the same as in the earlier paper mentioned above. A new ingredient is the analysis of the limit of the Gauss sum power series. Indeed, passing to the limit of the Gauss sum power series when \(n\to\infty\), the author rediscovers Ihara's power series and Anderson's result. This gives a simpler proof to both Ihara's conjecture and Anderson's result.
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11T23
zbMATH DE Number: 809105
zbMATH Keywords: Ihara's conjecture
zbMATH Keywords: congruence on Gauss sums
zbMATH Keywords: Gauss sum power series
MaRDI profile type: Publication
full work available at URL: jnth.1995.1099
OpenAlex ID: W2068419620
DOI: 10.1006/JNTH.1995.1099
description: scientific article; zbMATH DE number 809105