Bloch groups, algebraic $K$-theory, units, and Nahm's Conjecture

Abstract

Given an element of the Bloch group of a number field~$F$ and a natural number~$n$, we construct an explicit unit in the field $F_n=F(e^{2 \pi i/n})$, well-defined up to $\nn$-th powers of nonzero elements of~$F_n$. The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of~$F$ with the $K$-group $K_3(F)$ gives \changed{(up to an unidentified invertible scalar)} a \changed{formula} for a certain abstract Chern class from~$K_3(F)$. The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove Nahm's conjecture relating the modularity of certain $q$-hypergeometric series to the vanishing of the associated elements in the Bloch group of~$\overline{\mathbb Q}$.