Construction of all polynomial relations among Dedekind eta functions of level \(N\) (Q2000261)

The aim of this paper is to provide an algorithm to compute a Gröbner basis for the relations among Dedekind \(\eta\)-functions. Let \(\mathbb{H} = \{c\in\mathbb{C} \mid \Im(c)> 0 \}\) denote the complex upper half-plane and \[ \eta: \mathbb{H} \to \mathbb{C},\ \eta(\tau)=\exp(\pi i\tau/12)\prod_{k=1}^{\infty}(1-\exp(2\pi ik \tau))) \] Let us fix a positive integer \(N\) and let \(1 = \delta_1 < \delta_2 < \dots < \delta_n = N\) be the positive divisors of \(N\). We define \(\phi : \mathbb{Q}[E_{\delta_1} , \dots , E_{\delta_n}]\to \mathbb{Q}[\eta(\delta_1 \tau),\dots,\eta(\delta_n \tau)]\) with \(\phi(E_i)=\eta(\delta_i \tau)\) for each \(i\) where \(E_{\delta_1} , \dots , E_{\delta_n}\) is a sequence of variables. The main contribution of the paper is to compute a Gröbner basis for the kernel of \(\phi\).