On the calculation of certain Hilbert norm residue symbols and its application
Let \(\ell\) be an odd prime number, and let \(\mathbb{Z}_\ell\) and \(\mathbb{Q}_\ell\) be the ring of \(\ell\)-adic integers and the field of \(\ell\)-adic numbers, respectively. Fix an algebraic closure \(\overline \mathbb{Q}_\ell\) of \(\mathbb{Q}_\ell\) and a primitive \(\ell^n\)-th root of unity \(\zeta_{\ell ^n}\) in \(\overline \mathbb{Q}_\ell\) such that \(\zeta^\ell_{\ell^{n + 1}} = \zeta_{\ell^n}\) for each integer \(n \geq 1\). Let \((\alpha, \beta)_n\) denote the Hilbert norm residue symbol in \(\mathbb{Q}_\ell (\zeta_{\ell^n})\) for the power \(\ell^n\). Let \([\alpha, \beta]_n\) be an element in \(\mathbb{Z}/ \ell^n \mathbb{Z}\) defined by \((\alpha, \beta)_n = \zeta_{\ell^n}^{[\alpha, \beta]_n}\). Further, for each integer \(a\), let \(\xi_a = \exp (- \sum^\infty_{{i = 1 \atop \ell \nmid i}} {(\pi^a_n)^i \over i})\) be the \(\ell \)-adic exponential function with \(\pi_n = 1 - \zeta_{\ell^n}\). In this paper, the Hilbert residue symbols \([\xi_a, 1 + \ell^n]_n\) and \([\xi_a, \ell^n]_n\) for each integer \(a > \ell^{n - 1}\) are calculated, using Iwasawa's explicit formula for norm residue symbols. Theorem: (a) If \(a > \ell ^{n - 1}\), then \[ [\xi_a,1 + \ell^n]_n \equiv \begin{cases} - i \ell^{n - 1} \pmod {\ell^n} \quad & \text{if } \ell^{n - 1} + \ell^{n - 1 - i} \leq a < \ell^{n - 1} + \ell^{n - i} \\ 0 \pmod {\ell^n} \quad & \text{if } a \geq 2 \ell^{n - 1} \end{cases}. \] (Here \(i\) runs over integers \(1 \leq i \leq n - 1\).) (b) With \(\xi_a \) as above \[ [\xi_a, \ell^n]_n \equiv \begin{cases} - \ell^{n - 1} \pmod {\ell^n} \quad & \text{if } \ell^{n - 1} < a < 2 \ell^{n - 1} \\ 0 \pmod {\ell^n} \quad & \text{if } a \geq 2 \ell^{n - 1} \end{cases}. \] Corollary: (a) The conductor of the character \(\alpha \mapsto (\alpha, 1 + \ell^n)_n\) is \((\pi^2_1) \). (b) For \(n \geq 2\), the conductor of the character \(\alpha \mapsto (\alpha, \ell^n)_n\) is \((\pi^2_1)\). The proof is based on Iwasawa's explicit formulas for norm residue symbols [\textit{K. Iwasawa}, J. Math. Soc. Japan 20, 151-165 (1968; Zbl 0256.12013)]. As an application, a new proof is given for a theorem of Coleman and McCallum [\textit{R. Coleman} and \textit{W. McCallum}, J. Reine Angew. Math. 385, 41-101 (1988; Zbl 0654.12003)] about the conductor of the character \(\alpha \mapsto (\alpha, \ell^g (1 + \ell)^h)_n\) for \(g \in \mathbb{Z}\), \(h \in \mathbb{Z}_\ell\).
scientific article; zbMATH DE number 714111