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The \(p\)-adic Kummer-Leopoldt constant: normalized \(p\)-adic regulator. (English) Zbl 1393.11071

Let \(K\) be a number field and \(p\) a prime integer. Let \(\mathfrak{p}\) be a prime of \(K\) above \(p\), denote by \(E_K\) the group of \(p\)-principal global units of \(K\), that are units \(\varepsilon\equiv1 \pmod{\prod_{\mathfrak{p}|p}\mathfrak{p}}\). Let \(K_{\mathfrak{p}}\) be the \(\mathfrak{p}\)-completion of \(K\) and \(\mathfrak{I}_{\mathfrak{p}}\) the prime ideal corresponding to its ring of integers; denote by \(U_K\) the \(Z_p\)-module of principal local units at \(p\), that are elements of \(\bigoplus_{\mathfrak{p}|p} K_{\mathfrak{p}}^{\times} \) such that \(u=1+x\) where \(x\in \bigoplus_{\mathfrak{p}|p}\mathfrak{I}_{\mathfrak{p}}\). Finally, let \(\overline{E}_K\) denote the topological closure of \(E_K\) in \(U_K\). The Leopoldt conjecture states that the \(p\)-adic regulator of any number field \(K\) is \(\neq0\), which is equivalent to the condition that \(\mathrm{rank}_{Z_p}(\overline{E}_K)=\mathrm{rank}_{Z}(E_K)\) [L. C. Washington, Introduction to cyclotomic fields. 2nd ed. New York, NY: Springer (1997; Zbl 0966.11047)].
For a number field \(K\) the Kummer-Leopoldt constant \(\kappa\), if it exists, is the smallest integer \(c\) such that: for all \(n \gg 0\), if a unit \(\varepsilon\in E_K\) is in \(U_{K}^{p^{n+c}}\), then it is in \( E_K^{p^n}\). Note that if the Leopoldt’s conjecture at \(p\) is satisfied, then \(\kappa\) exists. This constant has been studied in many papers: [J. Assim and T. Nguyen Quang Do, Manuscr. Math. 115, No. 1, 55–72 (2004; Zbl 1081.11071); B. Anglès, Acta Arith. 98, No. 1, 33–51 (2001; Zbl 1002.11076); F. Lorenz, St. Petersbg. Math. J. 10, No. 6, 1 (1998; Zbl 0936.11063); M. Ozaki, Acta Arith. 81, No. 1, 37–44 (1997; Zbl 0873.11059)] and in some others papers.
In the present paper, the author gives an elementary \(p\)-adic proof and an improvement of some results about this constant and interprets it by using class field theory. He ends the paper by some applications, like as generalizations of Kummer’s lemma on regular \(p\)th cyclotomic fields, and a natural definition of the normalized \(p\)-adic regulator for any field \(K\) and any \(p \geq2\) by using class field theory and the properties of the \(p\)-torsion group of abelian \(p\)-ramification theory over \(K\).

MSC:

11R27 Units and factorization
11R37 Class field theory
11R29 Class numbers, class groups, discriminants
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References:

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