Nahm's conjecture: asymptotic computations and counterexamples (Q432999)

Werner Nahm came up with the following conjecture:NEWLINENEWLINEFor a positive definite symmetric \(r \times r\) matrix with rational coefficients \(A\), the following are equivalent: (i) The elements \([Q_1]+ \ldots +[Q_r]\) is torsion in the corresponding Bloch group for every solution of \(1 - Q_i = \prod_{j =1}^{r} Q_{j}^{A_ij}\), where \(i = 1, \dots,r\), \(A\) a real positive definite symmetric \(r \times r\) matrix. (ii) There exist \(B \in \mathbb{Q}^{r}\) and \(C \in \mathbb{Q}\) such that the followig series is a modular function: NEWLINE\[NEWLINEF_{A,B,C}(q) = \sum_{n = (n_1, \dots, n_r) \in (\mathbb{Z}_{\geq 0})^r}\frac {q^{1/2n^T An+n^TB+C}} {(q)_n1 \ldots (q)_nr},NEWLINE\]NEWLINE where \(r \geq 1\) is a positive integer, \(A\) a real positive definite symmetric \(r \times r\)-matrix, \(B\) a vector of length \(r\) and \(C\) a scalar.NEWLINENEWLINEIn this paper, the author provides counterexamples to show that condition (ii) does not imply condition (i). The other way round, i.e., from (i) to (ii), could be true but we do not know it yet. As the author mentions, the correct formulation of the conjecture remains an interesting open question.