The syntomic regulator for the K-theory of fields

Abstract

We define complexes analogous to Goncharov's complexes for the K-theory of discrete valuation rings of characteristic zero. Under suitable assumptions in K-theory, there is a map from the cohomology of those complexes to the K-theory of the ring. In case the ring is the localization of the ring of integers in a number field, there are no assumptions necessary. We compute the composition of our map to the K-theory with the syntomic regulator. The result can be described in terms of a p-adic polylogarithm. Finally, we apply our theory in order to compute the regulator to syntomic cohomology on Beilinson's cyclotomic elements. The result is again given by the p-adic polylogarithm. This last result is related to one by Somekawa and generalizes work

On the p-adic Beilinson Conjecture for Number Fields

“…In [5], two of the authors of the present work showed how one can sometimes compute the map reg p by using p-adic polylogarithms.…”

“…Thus it provides a way of computing them, and Paul Buckingham wrote a computer implementation for this. In [5] Besser and de Jeu computed the syntomic regulator for (essentially) the part of the K-theory of a number field described by Zagier's conjecture and showed that it is given by applying the p-adic polylogarithm. Those p-adic polylogarithms were invented by Coleman [16] using his theory of p-adic integration but are not so easy to compute.…”

“…cit. (up to sign since one has to use the correct formula for ω n , which can be obtained as in [5,Section 6]). Using the computations on pages 236 and 237 of [19] one then sees that Π ∞ maps ε n (z) to (−1) n (n − 1)!P n (z), where P n (z) = π n−1…”

On the syntomic regulator for K 1 of a surface

“…Formulas for the syntomic regulator are to be thought of as p-adic analogues of formulas for the Beilinson regulator [Bei85]. Several such formulas exists [Bes00c,BdJ03,BdJ04] for zero and one-dimensional varieties.…”

The syntomic regulator forK4of curves

“…Construction of the complexes for F and C . Several parts of the constructions of the complexes in this section and in Section 2.5 below were carried out in earlier papers [de Jeu 1995;1996;Besser and de Jeu 2003], but we review them so that we can refer to the relevant details in some new constructions for ᏻ and in the calculations relating to regulators in later sections. Also, in various cases the constructions were carried out more generally, in which case they tend to become dependent on assumptions on weights in K-theory, and our exposition below will avoid such assumptions.…”