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Syntomic regulators and $$p$$-adic integration. II: $$K_2$$ of curves. (English) Zbl 1001.19004
Let $$C$$ be a smooth complete curve over $$\mathbb{C}_p$$ with good reduction. R. Coleman and E. deShalit [Invent. Math. 93, No. 2, 239-266 (1988; Zbl 0655.14010)] defined a $$p$$-adic regulator $r_{p,C}: K_2(\mathbb{C}_p(C)) \rightarrow \text{Hom} (H^0(C,\Omega^1_{C/\mathbb{C}_p}),\mathbb{C}_p)$ based on Coleman’s $$p$$-adic integration theory. In the case that $$C$$ arises via base change from a smooth surjective scheme $$Z$$ over the ring of integers in a finite extension of $$\mathbb{Q}_p$$, the author compares the regulator of Coleman and deShalit with the syntomic regulator $c_2^p:K_2(Z) \rightarrow H^{2}_{\text{syn}}(Z,2)$ defined by the author in the first part of this paper [A. Besser, Isr. J. Math. 120, Pt. B, 291-334 (2000; Zbl 1001.19003)]. The main result shows that these two regulators are in fact canonically related via Poincaré duality. The main technical tool is the notion of a local index, a generalized residue. The main result then follows from a generalization of R. F. Coleman’s reciprocity law [Compos. Math. 72, No. 2, 205-235 (1989; Zbl 0706.14013)] in terms of Coleman functions and local indices.

##### MSC:
 19E20 Relations of $$K$$-theory with cohomology theories 14F30 $$p$$-adic cohomology, crystalline cohomology
##### Citations:
Zbl 0655.14010; Zbl 1001.19003; Zbl 0706.14013
Full Text:
##### References:
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