Multiplicative differential algebraic $K$-theory and applications

Abstract

We construct a version of Beilinson’s regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic $K$-theory. We use this theory to give an interpretation of Bloch’s construction of $K_3$-classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic $K$-theory of number rings.