  We consider certain q-series depending on parameters (A,B,C), where A is a
positive definite r times r matrix, B is an r-vector and C is a scalar, and ask
when these q-series are modular forms. Werner Nahm conjectured a criterion for
which A's can occur, in terms of torsion in the Bloch group. The conjecture was
proved by Don Zagier and Michael Terhoeven for r=1. We develop their approach
for r>1 and find several new examples of modular forms as well as some
counterexamples to Nahm's conjecture.
