A note on the universal power series for Jacobi sums
Let \(\ell\) be an odd prime. \textit{Y. Ihara}, \textit{M. Kaneko} and \textit{A. Yukinari} [Adv. Stud. Pure Math. 12, 65-86 (1987; Zbl 0642.12012)] constructed a power series related to Jacobi sums, from which one obtains a homomorphism \(\tilde g:\) Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}}(\mu_{\ell^{\infty}}))\to {\mathbb{F}}_{\ell}t\). Let \(\tilde {\mathcal V}^-\) denote the additive group of power series \(g(t)\in {\mathbb{F}}_{\ell}t\) satisfying \(g((1+t)^{-1}-1)=-g(t)\) and \(D^{\ell -1}g=g\), where \(D=(1+t)d/dt\). In [J. Number Theory 31, 312-334 (1989; Zbl 0686.12003)] the author and \textit{M. Kaneko} showed that Im \(\tilde g\subseteq \tilde {\mathcal V}^-\), with equality if and only if Vandiver's conjecture holds for \(\ell\). In the present paper it is shown that \(\tilde {\mathcal V}^-/Im \tilde g\) has cardinality \(\ell^{\lambda +\epsilon}\), where \(\lambda\) is the Iwasawa invariant for the cyclotomic \({\mathbb{Z}}_{\ell}\)-extension of \({\mathbb{Q}}(\cos (2\pi /\ell))\) and \(\epsilon\) is a certain invariant defined in terms of the maximal abelian \(\ell\)-extension of \({\mathbb{Q}}(\mu_{\ell^{\infty}})\) unramified outside \(\ell\).
scientific article; zbMATH DE number 4158758
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