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
-theory.
We use this theory to give an interpretation of Bloch’s construction of
-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
-theory of
number rings.