## 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)

### Citations:

Zbl 0446.00004; Zbl 0437.12008
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