Log-syntomic regulators and \(p\)-adic polylogarithms.

*(English)*Zbl 0978.19004The 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.

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.

Reviewer: Jean-François Jaulent (Talence)

##### 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 |