An elementary proof of Newman's eta-quotient theorem

Abstract

Let eta(z) be the Dedekind eta function. Newman studied the modularity of eta-quotients, giving necessary and sufficient conditions for a function of the form \prod_{0 < m | N} eta(mz)^{r_m} to be a (weakly) holomorphic modular form of level N. We explain a proof of Newman's theorem, developed while teaching a class for talented high school students at Canada/USA Mathcamp. The key observation is that although Gamma_1(N) is not generated by its upper triangular and lower triangular subgroups, it is generated by those subgroups together with any congruence subgroup. Modularity with respect to some congruence subgroup is established using one simple identity involving the multiplier system of eta(z), whose proof is elementary in the sense that it avoids the use of Dedekind sums.