Ideal of Relations of eta Functions

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

The relations are expressed as polynomials in variables \(E1, E2\), etc. where for a divisor \(δ\) of a given level \(N\) the variable \(Eδ\) stands for the eta function \(η(δτ)\).

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

with \(q = q(τ) = \exp(2πiτ)\).

The relations listed by Somos can easily be expressed in terms of the \(Eδ\) variables (i.e. directly in terms of eta functions). These (transformed) relations can be represented in terms of the Gröbner bases from this site. We give a collection of such representations for various entries in the collection of Somos.

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.

All files can be read directly by FriCAS, in particular, reading the fricassomos*.input files evaluates the representation of relations from Somos’ list in terms of the Gröbner basis of relations and thus yields a list of zeros. See EtaRelations8 for an example of evaluation in FriCAS and SandboxSomos2Eta for a FriCAS implementation to translate the notation of Somos into our notation.

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.