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Log-syntomic regulators and $$p$$-adic polylogarithms. (English) Zbl 0978.19004
The author computes the image of the Beilinson element under the syntomic regulator map from the motivic cohomology to the syntomic cohomology of the integer ring of a cyclotomic field in terms of values of the $$p$$-adic polylogarithm.
More precisely, let $$\zeta_n$$ be a primitive $$n$$-th root of unity and $$O_F = \mathbb{Z}[\zeta_n]$$ be the ring of integers of the $$n$$-th cyclotomic field $$F$$. For any $$r \geq 2$$, A. A. Beilinson [J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984; Zbl 0588.14013)], by using Loday symbol in $$K$$-theory, constructed a cyclotomic element $$C^B_r(\zeta_n)$$ in the motivic cohomology $$H^1_\mathcal M (O_F, \mathbb{Q}(r)) = \mathbb{Q} \otimes K_{2r-1}(O_F)$$ and proved that the image of this element under the regulator map to Deligne cohomology $$H^1_ \mathcal D(F_\mathbb{R}, \mathbb{R}(r))$$ is represented by special values of the r-th polylogarithmic function at $$\zeta_n$$. Partial $$p$$-adic analogues of this result for prime $$p \nmid n$$ were given for $$r =2$$ first by R. F. Coleman by using rigid analytic calculations of the $$p$$-adic dilogarithm [Invent. Math. 69, 171-208 (1982; Zbl 0516.12017)] and next by M. Gros by using the syntomic cohomology [Invent. Math. 99, 293-320 (1990; Zbl 0667.14006)]. Furthermore R. Gross extended these results to $$r \geq 2$$ by using the rigid syntomic regulator [Invent. Math. 115, No. 1, 61-79 (1994; Zbl 0799.14010)].
The goal of the paper under review is to generalize these previous computations without the restriction $$p \nmid n$$. The author works with an other construction of the $$C^B_r(\zeta_n)$$, also due to A. A. Beilinson [“Polylogarithm and cyclotomic elements”, Preprint (1989; unreviewed)]. In the first part of the article, he defines log-syntomic cohomologies for regular syntomic schemes with relative simple normal crossing divisors; thus in the second part he develops a theory of regulators with values in the syntomic cohomology with logarithmic poles along horizontal divisors involving Chern class maps of log-syntomic complexes; finally he introduces the $$p$$-adic polylogarithm and applies the above construction to the simplicial schemes considered by Beilinson.
The exposition of the paper is very concise but in the opinion of the reviewer a bit elliptic since many technical details are unfortunately left to the reader.

##### MSC:
 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 14F99 (Co)homology theory in algebraic geometry 19E99 $$K$$-theory in geometry
##### Citations:
Zbl 0588.14013; Zbl 0516.12017; Zbl 0667.14006; Zbl 0799.14010
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