Infinitesimal Chow dilogarithm

Abstract

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 ?(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 [Algebra Number Theory 3 (2009), pp. 1.34]. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture of Goncharov [J. Amer. Math. Soc. 18 (2005), pp. 1.60] with an explicit formula for the homotopy map. Also using ?, we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus [Amer. J. Math. 131 (2009), pp. 257-276].