On Ihara's power series
\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.
scientific article; zbMATH DE number 809105