### abstract

- Let C be a curve dened over a discrete valuation eld of char- acteristic zero where the residue eld has positive characteristic. Assuming that C has good reduction over the residue eld, we compute the syntomic regulator on a certain part of K (3) 4 (C). The result can be expressed in terms of p-adic polylogarithms, and Coleman integration. j K(j) n (X) ; where K (j) n (X) consists of those in Kn(X) Z Q such that k ( ) = kj for all Adams operators k, see (Sou85, Proposition 5). There is a regulator map K(j) n (X)! H 2j n syn (X;j); see (Bes00a). In many interesting cases the target group of the regulator is isomor- phic to the rigid cohomology group of the special b er X , in the sense of Berthelot, H2j n 1 rig (X =K). We will be interested in the situation where X is a proper smooth curve C over R with generic b er C, and the K-group is K (3)