Infinitesimal Chow Dilogarithm

TL;DR

This paper introduces an infinitesimal invariant for algebraic cycles on a smooth curve over dual numbers, extending the Chow dilogarithm and proving an infinitesimal reciprocity conjecture.

Contribution

It defines a new invariant extending the Chow dilogarithm to infinitesimal settings and proves an associated reciprocity conjecture.

Findings

Established an infinitesimal version of the strong reciprocity conjecture.

Constructed an infinitesimal regulator for algebraic cycles of weight two.

Extended Park's regulator construction to cycles with modulus.

Abstract

Let $C_{2}$ be a smooth and projective curve over the ring of dual numbers of a field $k.$ Given non-zero rational functions $f,g,$ and $h$ on $C_{2},$ we define an invariant $\rho(f\wedge g \wedge h) \in k.$ This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture. Also using $\rho,$ we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus.