Infinitesimal Chow dilogarithm

Let C2 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 C2, we define an invariant ρ(f∧g∧h) ∈ k. This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm of [11]. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture [5]. Also using ρ, we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park’s construction in the case of cycles with modulus [8].