Deformations of Bloch groups and Aomoto dilogarithms in characteristic p

In this paper, we study the Bloch group B2(F[ε]2) over the ring of dual numbers of the algebraic closure of the field with p elements, for a prime p⩾5. We show that a slight modification of Kontsevichʼs 112-logarithm defines a function on B2(F[ε]2). Using this function and the characteristic p version of the additive dilogarithm function that we previously defined, we determine the structure of the infinitesimal part of B2(F[ε]2) completely. This enables us to define invariants on the group of deformations of Aomoto dilogarithms and determine its structure. This final result might be viewed as the analog of Hilbertʼs third problem in characteristic p.