  Let $C$ be a curve defined over a discrete valuation field of characteristic
zero where the residue field has positive characteristic. Assuming that $C$ has
good reduction over the residue field, we compute the syntomic regulator on a
certain part of $K_4^[(3)}(C)$. The result can be expressed in terms of
$p$-adic polylogarithms and Coleman integration. We also compute the syntomic
regulator on a certain part of $K_4^[(3)}(F)$ for the function field $F$ of
$C$. The result can be expressed in terms of $p$-adic polylogarithms and
Coleman integration, or by using a trilinear map ("triple index") on certain
functions.
