The Stickelberger elements and the cyclotomic units in the cyclotomic \(\mathbb{Z}_p\)-extensions (Q2757134)

The \(p\)-adic \(L\)-functions of Kubota and Leopoldt were originally defined by \(p\)-adic interpolation of complex \(L\)-functions. As is well known, \textit{K. Iwasawa} [Lectures on \(p\)-adic \(L\)-functions, Ann. Math. Stud. 74 (1972; Zbl 0236.12001)] gave a second, more arithmetical construction based on norm coherent sequences of Stickelberger elements in cyclotomic \(\mathbb Z_p\)-extensions of cyclotomic fields. One key ingredient is that the value of a Dirichlet character \(\chi\) applied to such a sequence is essentially a special value of \(L(s,\chi)\) up to certain Euler factors. As special values of \(p\)-adic \(L\)-functions can also be expressed in terms of \(p\)-adic logarithms of cyclotomic units [Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, Springer (1997; Zbl 0966.11047), Theorem 5.18] (a ``not especially enlightening proof'', \textit{L. Washington} dixit), one asks naturally for a functorial relationship between cyclotomic units and Stickelberger elements. A functorial framework is provided by the Coleman maps NEWLINE\[NEWLINE\Psi_N: \lim(\mathbb Z[\zeta_{Np^{n+1}}]\otimes\mathbb Z_p)^\times \to \lim(\mathbb Z_p [\zeta_N])[\text{Gal} (\mathbb Q(\zeta_{p^{n+1}})/\mathbb Q)],NEWLINE\]NEWLINE and the theorem of Iwasawa-Coleman [\textit{R. Coleman}, Proc. Am. Math. Soc. 89, 1-7 (1983; Zbl 0528.12005), Proposition 6] gives a precise formula relating \(\Psi_1(\eta_1)\) to \(\Theta_1^*\), where \(\eta_1\) (resp. \(\Theta_1\)) is a norm coherent sequence of cyclotomic units (resp. Stickelberger elements) and the star means taking the inverse Galois action. The present paper generalizes this result from \(N=1\) to higher \(N\), prime to \(p\). One key technical step consists in using Leopoldt's ``Basiszahl'' to identify the target of \(\Psi_N\) with the completed group ring \(\mathbb Z_p[[\text{ Gal}(\mathbb Q(\zeta_{Np^\infty})/\mathbb Q)]]\). NEWLINENEWLINENEWLINEThe main result (Theorem 2.1) then gives the image by \(\Psi_N\) of \(\eta_N\), as well as a pre-image of a modified \(\Theta_N^*\) (with obvious notation). The author gives two proofs: the first one based on the expression of \(\Psi_N\) by means of logarithmic derivatives of Coleman's power series; the second one following Iwasawa and Coleman's steps (op. cit.).