Eta Function Relations by Somos

Below we relate the (degrevlex) Gröbner bases of relations among eta functions for various levels with the list of such relations computed by Michael Somos.

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 listed by Somos can easily be expressed in terms of the \(e_\delta\) variables (i.e. directly in terms of eta functions) by means of the following translation code from QEta:

somos := 1*u1^8*u4^4 +8*q*u1^4*u2^2*u4^2*u8^4 -1*u2^12
numer rationalFunction etaExpression qpSPEX(QQ)(somos, "u")

From eta notation to math:q-Pochhammer notation one can use this code:

erel := 8*e1^4*e2^2*e4^2*e8^4+e1^8*e4^4-e2^12
rationalFunction(qExpression eqSPEX(ZZ)(erel))

The result agrees with the expression of Somos up to a factor of \(q\) and renaming of the variables.

These (transformed) relations can be represented in terms of the Gröbner bases from Ideal of Relations of eta Functions.

The collection is also in the QEta-Data repository.

There are 3 sections in each file:

  • eta relations — lists the elements of the degrevlex Gröbner bases and gives names to those polynomials,

  • Somos eta relations — list of elements from Somos’ list with the same identifier, but translated from \(q\)-Pochhammer notation into eta-notation,

  • Relation relations — relates Somos’ elements to the Gröbner basis so that the resulting polynomial will give 0 when the Somos elements and Gröbner basis elements are replaced by their respective expression in eta-functions.

List of Somos and its connection to Gröbner bases of Relations of eta Functions