Dedekind eta function
More generally, suppose a, b, c, d are integers with ad − bc = 1, so that is a transformation belonging to the modular group.
The picture on this page shows the modulus of the Euler function: the additional factor of q1/24 between this and eta makes almost no visual difference whatsoever.
The characters themselves allow the construction of generalizations of the Jacobi theta function which transform under the modular group; this is what leads to the identities.
Eta quotients may also be a useful tool for describing bases of modular forms, which are notoriously difficult to compute and express directly.
In 1993 Basil Gordon and Kim Hughes proved that if an eta quotient ηg of the form given above, namely
satisfies then ηg is a weight k modular form for the congruence subgroup Γ0(N) (up to holomorphicity) where[4] This result was extended in 2019 such that the converse holds for cases when N is coprime to 6, and it remains open that the original theorem is sharp for all integers N.[5] This also extends to state that any modular eta quotient for any level n congruence subgroup must also be a modular form for the group Γ(N).
While these theorems characterize modular eta quotients, the condition of holomorphicity must be checked separately using a theorem that emerged from the work of Gérard Ligozat[6] and Yves Martin:[7] If ηg is an eta quotient satisfying the above conditions for the integer N and c and d are coprime integers, then the order of vanishing at the cusp c/d relative to Γ0(N) is These theorems provide an effective means of creating holomorphic modular eta quotients, however this may not be sufficient to construct a basis for a vector space of modular forms and cusp forms.
For example, if we assume N = pq is a semiprime then the following process can be used to compute an eta-quotient basis of Mk(Γ0(N)).
[5] and by noticing that the sum of the orders of vanishing at the cusps of Γ0(N) must equal A collection of over 6300 product identities for the Dedekind Eta Function in a canonical, standardized form is available at the Wayback machine[9] of Michael Somos' website.
