Infinitesimal Chow Dilogarithm

Let $C_2$ be a smooth and projective curve over the ring of dual numbers of a field $k\mathrm{.}$ Given non-zero rational functions $f,g,$ and $h$ on $C_2,$ we define an invariant $\rho (f\wedge g\wedge h)\in k\mathrm{.}$ 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.