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.


19E20 Relations of \(K\)-theory with cohomology theories
14F30 \(p\)-adic cohomology, crystalline cohomology
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[1] A. Besser and R. de Jeu,The syntomic regulator for K-theory of fields, Work in progress, 1999. · Zbl 1106.11024
[2] A. A. Beilinson,Higher regulators and values of L-functions of curves, Functional Analysis and its Applications14 (1980), 116-118. · Zbl 0475.14015 · doi:10.1007/BF01086554
[3] P. Berthelot,Cohomologie rigide et cohomologie rigide a support propres, premier partie, Preprint 96-03 of the University of Rennes, available at http://www.maths.univ-rennes1.fr/ berthelo/, 1996. · Zbl 0655.14010
[4] P. Berthelot,Finitude et pureté cohomologique en cohomologie rigide, Inventiones Mathematicae128 (1997), 329-377, with an appendix in English by A. J. de Jong. · Zbl 0908.14005 · doi:10.1007/s002220050143
[5] A. Besser,Syntomic regulators and p-adic integration I: rigid syntomic regulators, this volume. · Zbl 1001.19003
[6] R. Coleman and E. de Shalit,p-adic regulators on curves and special values of p-adic L-functions, Inventiones Mathematicae93 (1988), 239-266. · Zbl 0655.14010 · doi:10.1007/BF01394332
[7] R. Coleman,Reciprocity laws on curves, Compositio Mathematica72 (1989), 205-235. · Zbl 0706.14013
[8] M. Gros,Régulateurs syntomiques et valeurs de fonctions L p-adiques. I, Inventiones Mathematicae99 (1990), 293-320, with an appendix by Masato Kurihara.. · Zbl 0667.14006 · doi:10.1007/BF01234421
[9] G. Harder,Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern, Inventiones Mathematicae42 (1977), 135-175. · Zbl 0391.20036 · doi:10.1007/BF01389786
[10] N. Katz and W. Messing,Some consequences of the Riemann hypothesis for varieties over finite fields, Inventiones Mathematicae23 (1974), 73-77. · Zbl 0275.14011 · doi:10.1007/BF01405203
[11] W. Nizioł,On the image of p-adic regulators, Inventiones Mathematicae127 (1997), 375-400. · Zbl 0928.14014 · doi:10.1007/s002220050125
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