Nahm's conjecture on Bloch-group torsion and modularity

Let $A$ be a positive definite symmetric $r\times r$ matrix with rational coefficients. Let $(Q_1,\dots,Q_r)$ range over the solutions in number fields of the algebraic system

, and let $F_{A,B,C}$ denote the function associated with $A$, $B$, and $C$ in the paper. The corresponding Bloch group is the quotient of the kernel of

\mathbb Z[K^\setminus 1]\longrightarrow \Lambda^2 K^,\qquad [x]\longmapsto x\wedge(1-x),

by the five-term relations described above. **Nahm's conjecture.** The following are equivalent: (i) for every solution $(Q_1,\dots,Q_r)$ of

, the element

$$[Q_1]+\dots+[Q_r]$$

is torsion in the corresponding Bloch group; and (ii) there exist $B\in\mathbb Q^r$ and $C\in\mathbb Q$ such that $F_{A,B,C}$ is a modular function. The conjecture links the torsion condition detected by the Bloch-group regulator and dilogarithm identities with modularity of the associated Nahm sum.