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On higher \(p\)-adic regulators. (English) Zbl 0488.12008

Algebraic \(K\)-theory, Proc. Conf., Evanston 1980, Lect. Notes Math. 854, 372-401 (1981).
Let \(F\) be a totally real abelian number field with ring of integers \(\mathfrak o_F\), \(p\) an odd prime not dividing the degree of \(F\) over \(\mathbb Q\) and \(F_p\), the product of the completions of \(F\) at the places dividing \(p\). Among others the author proves the following:
If \(i\ge 3\) is an odd integer and if the Kubota-Leopoldt \(p\)-adic \(L\)-function \(L_p(F,\omega^{i-1},s)\) is non-zero at \(s=i\), the morphism \[K_{2i-1}(\mathfrak o_F)/\text{torsion} \to K_{2i-1}(F_p)/\text{torsion} \] is injective. The proof uses the comparison between \(K\)-theory and étale cohomology for global and local fields developed by the author in [Invent. Math. 55, 251–295 (1979; Zbl 0437.12008)].
[For the entire collection see Zbl 0446.00004.]

MSC:

11R70 \(K\)-theory of global fields
11R23 Iwasawa theory
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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