Ideal of Relations of eta Functions

Below we list the polynomials of a (degrevlex) Gröbner basis of relations among eta functions for various levels.

This data is also available in the QEta-Data git repository.

The relations are expressed as polynomials in variables \(e1, e2\), etc. where for a divisor \(\delta\) of a given level \(N\) the variable \(e_\delta\) stands for the eta function \(\eta(\delta\tau)\).

\[η(δτ) = \exp\left(\frac{πiτ}{12}\right) \prod_{n=1}^\infty (1 − q^{δn})\]

with \(q = q(τ) = \exp(2πiτ)\).

The relations have been computed by an implementation of the algorithm samba from “Dancing Samba with Ramanujan Partition Congruences” in the computer algebra system FriCAS and the slimgb (Gröbner bases) algorithm from the system Singular.

The relations listed by Somos can easily be expressed in terms of the \(e_\delta\) variables (i.e. directly in terms of eta functions). These (transformed) relations can be represented in terms of the Gröbner bases from here. We give a collection (related website: Eta Function Relations by Somos) of such representations for various entries in the collection of Somos.

Details of how exactly the relations have been computed is explained in the article “Construction of all Polynomial Relations among Dedekind Eta Functions of Level N” by Hemmecke and Radu in the Journal of Symbolic Computation. The article is also available as RISC report 18-03.

List of Gröbner bases of Relations of eta Functions