Infinitesimal dilogarithm on curves over truncated polynomial rings

Abstract

Let $C$ be a smooth and projective curve over the truncated polynomial ring $k_m:=k[t]/(t^m), $ where $k$ is a field of characteristic 0. Using a candidate for the motivic cohomology group ${\rm H}^{3}_{\pazocal{M}}(C,\mathbb{Q}(3))$ based on the Bloch complex of weight 3, we construct regulators to $k$ for every $m<r<2m.$ Specializing this construction, we obtain an invariant $\rho_{m,r}(f \wedge g \wedge h)$ of rational functions $f,$ $g$ and $h$ on $C.$ The current work is a twofold generalization of our work on the infinitesimal Chow dilogarithm: we sheafify the previous construction and therefore do not restrict ourselves to triples of rational functions and we construct the regulator for any $m<r<2m,$ rather than only for $m=2.$ We also define regulators of cycles, which we expect to give a complete set of invariants for the infinitesimal part of ${\rm CH}^{2}(k_{m},3). $ This generalizes Park's work, where the additive Chow cycles, namely the case of cycles close to 0, is handled for $r=m+1.$ In this paper, we generalize the reciprocity theorem to pairs of cycles which are the same modulo $(t^m)$ and for any $m<r<2m.$ We expect the theory of the paper to give regulators on categories of motives over rings with nilpotents.