Nahm's conjecture

We consider certain $r$-fold $q$-hypergeometric series depending on several rational parameters. These series arise in conformal field theory and it is of interest to know for which values of parameters they are modular. A conjectural (partial) answer by Werner Nahm surprisingly involves dilogarithms and the Bloch group. This conjecture was proved by Don Zagier for rank $r=1$ and also tested numerically by him and Michael Terhoeven for higher ranks $r>1$. We develop a general method of finding values of parameters corresponding to modular cases and apply it for $r=1$ (reproving Zagier's result) and $r=2$ (obtaining several modular cases not known earlier).