Editing Dedekind eta function (section)

==Eta quotients==
Eta quotients are defined by quotients of the form
:<math> \prod_{0<d\mid N}\eta(d\tau)^{r_d} </math>
where {{mvar|d}} is a non-negative integer and {{mvar|r<sub>d</sub>}} is any integer. Linear combinations of eta quotients at imaginary quadratic arguments may be [[algebraic number|algebraic]], while combinations of eta quotients may even be [[integer|integral]]. For example, define,

:<math>\begin{align}
j(\tau)&=\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{8}+2^8 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{16}\right)^3 \\[6pt]

j_{2A}(\tau)&=\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{12}+2^6 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{12}\right)^2 \\[6pt]

j_{3A}(\tau) &=\left(\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{6}+3^3 \left(\frac{\eta(3\tau)}{\eta(\tau)}\right)^{6}\right)^2 \\[6pt]

j_{4A}(\tau) &=\left(\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{4} + 4^2 \left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{4}\right)^2 = \left(\frac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \right)^{24} 
\end{align}</math>

with the 24th power of the [[Weber modular function]] {{math|𝔣(''τ'')}}. Then,

:<math>\begin{align}
j\left(\frac{1+\sqrt{-163}}{2}\right) &= -640320^3, & e^{\pi\sqrt{163}} &\approx 640320^3+743.99999999999925\dots \\[6pt]

j_{2A}\left(\frac{\sqrt{-58}}{2}\right) &= 396^4, & e^{\pi\sqrt{58}}&\approx 396^4-104.00000017\dots \\[6pt]

j_{3A}\left(\frac{1+\sqrt{-\frac{89}{3}}}{2}\right) &= -300^3, & e^{\pi\sqrt\frac{89}{3}}&\approx 300^3+41.999971\dots \\[6pt]

j_{4A}\left(\frac{\sqrt{-7}}{2}\right)&=2^{12}, & e^{\pi\sqrt{7}}&\approx 2^{12}-24.06\dots
\end{align}</math>

and so on, values which appear in [[Ramanujan–Sato series]].

Eta quotients may also be a useful tool for describing bases of [[modular form]]s, which are notoriously difficult to compute and express directly. In 1993 Basil Gordon and Kim Hughes proved that if an eta quotient {{mvar|η<sub>g</sub>}} of the form given above, namely <math> \prod_{0<d\mid N}\eta(d\tau)^{r_d} </math> satisfies
:<math> \sum_{0<d\mid N}d r_d \equiv 0 \pmod{24} \quad \text{and} \quad \sum_{0<d\mid N}\frac{N}{d}r_d \equiv 0 \pmod{24},</math>
then {{mvar|η<sub>g</sub>}} is a [[modular form|weight {{mvar|k}} modular form]] for the [[congruence subgroup]] {{math|Γ<sub>0</sub>(''N'')}} (up to [[Holomorphic function|holomorphicity]]) where<ref>{{cite book|first1=Basil |last1=Gordon |first2=Kim |last2=Hughes |contribution=Multiplicative properties of ''η''-products. II. |title=A Tribute to Emil Grosswald: Number Theory and Related Analysis |volume=143 |series=Contemporary Mathematics |page=415–430 |publisher=American Mathematical Society |location=Providence, RI |date=1993}}</ref> 
:<math>k=\frac12\sum_{0<d\mid N} r_d.</math>
This result was extended in 2019 such that the converse holds for cases when {{mvar|N}} is [[Coprime integers|coprime]] to 6, and it remains open that the original theorem is sharp for all integers {{mvar|N}}.<ref name="AAHOS">{{cite journal|first1=Michael |last1=Allen|first2=Nicholas |last2=Anderson|first3=Asimina |last3=Hamakiotes|first4=Ben |last4=Oltsik|first5=Holly |last5=Swisher|title=Eta-quotients of prime or semiprime level and elliptic curves|journal=Involve|year=2020|volume=13|issue=5|pages=879–900 |doi=10.2140/involve.2020.13.879|arxiv=1901.10511|s2cid=119620241 }}</ref>  This also extends to state that any [[modular form|modular eta quotient]] for any [[Congruence subgroup| level {{mvar|n}} congruence subgroup]] must also be a modular form for the group {{math|Γ(''N'')}}. While these theorems characterize [[Modular form|modular]] eta quotients, the condition of [[Holomorphic function| holomorphicity]] must be checked separately using a theorem that emerged from the work of Gérard Ligozat<ref>{{cite book|first=G. |last=Ligozat |title=Courbes modulaires de genre 1 |publisher=U.E.R. Mathématique, Université Paris XI, Orsay |date=1974 |series=Publications Mathématiques d'Orsay |volume=75 |page=7411}}</ref> and Yves Martin:<ref>{{cite journal|first=Yves |last=Martin|title=Multiplicative ''η''-quotients|journal=[[Transactions of the American Mathematical Society]]|year=1996|volume=348|issue=12|page=4825–4856|doi=10.1090/S0002-9947-96-01743-6 |doi-access=free}}</ref>

If {{mvar|η<sub>g</sub>}} is an eta quotient satisfying the above conditions for the integer {{mvar|N}} and {{mvar|c}} and {{mvar|d}} are coprime integers, then the order of vanishing at the [[Cusp (singularity)| cusp]] {{math|{{sfrac|''c''|''d''}}}} relative to {{math|Γ<sub>0</sub>(''N'')}} is 
:<math>\frac{N}{24}\sum_{0<\delta|N} \frac{\gcd\left(d,\delta\right)^2r_\delta}{\gcd\left(d,\frac{N}{\delta}\right)d\delta} .</math>
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 form|cusp forms]]. A useful theorem for limiting the number of modular eta quotients to consider states that a holomorphic weight {{mvar|k}} modular eta quotient on {{math|Γ<sub>0</sub>(''N'')}} must satisfy
:<math>\sum_{0<d\mid N} |r_d|\leq \prod_{p\mid N}\left(\frac{p+1}{p-1}\right)^{\min\bigl(2,\text{ord}_p(N)\bigr)},</math>
where {{math|ord<sub>''p''</sub>(''N'')}} denotes the largest integer {{mvar|m}} such that {{mvar|p<sup>m</sup>}} divides {{mvar|N}}.<ref name="RW">{{cite journal|first1=Jeremy |last1=Rouse|first2=John J. |last2=Webb|title=On spaces of modular forms spanned by eta-quotients|journal=[[Advances in Mathematics]]|year=2015|volume=272|page=200–224|doi=10.1016/j.aim.2014.12.002|doi-access=free|arxiv=1311.1460}}</ref>
These results lead to several characterizations of spaces of modular forms that can be spanned by modular eta quotients.<ref name="RW"/> Using the [[graded ring]] structure on the ring of modular forms, we can compute bases of vector spaces of modular forms composed of {{math|<math>\mathbb{C}</math>}}-linear combinations of eta-quotients. For example, if we assume {{math|''N'' {{=}} ''pq''}} is a [[semiprime]] then the following process can be used to compute an eta-quotient basis of [[modular form|{{math|''M<sub>k</sub>''(Γ<sub>0</sub>(''N''))}}]].<ref name="AAHOS" />

{{ordered list
|Fix a semiprime {{math|''N'' {{=}} ''pq''}} which is coprime to 6 (that is, {{math|''p'', ''q'' > 3}}). We know that any modular eta quotient may be found using the above theorems, therefore it is reasonable to algorithmically to compute them.

|Compute the dimension {{mvar|D}} of {{math|''M<sub>k</sub>''(Γ<sub>0</sub>(''N''))}}. This tells us how many linearly-independent modular eta quotients we will need to compute to form a basis.

|Reduce the number of eta quotients to consider. For semiprimes we can reduce the number of partitions using the bound on
:<math>\sum_{0<d\mid N} |r_d|</math> 
and by noticing that the sum of the orders of vanishing at the cusps of {{math|Γ<sub>0</sub>(''N'')}} must equal
:<math>S:=\frac{(p+1)(q+1)}{6}</math>.<ref name="AAHOS"/>

|Find all partitions of {{mvar|S}} into 4-tuples (there are 4 cusps of {{math|Γ<sub>0</sub>(''N'')}}), and among these consider only the partitions which satisfy Gordon and Hughes' conditions (we can convert orders of vanishing into exponents). Each of these partitions corresponds to a unique eta quotient.

|Determine the minimum number of terms in the [[modular form|{{mvar|q}}-expansion]] of each eta quotient required to identify elements uniquely (this uses a result known as [[Sturm's bound]]). Then use linear algebra to determine a maximal independent set among these eta quotients.

|Assuming that we have not already found {{mvar|D}} linearly independent eta quotients, find an appropriate vector space {{math|''M''<sub>''k''{{prime}}</sub>(Γ<sub>0</sub>(''N''))}} such that {{math|''k''{{prime}}}} and {{math|''M''<sub>''k''{{prime}}</sub>(Γ<sub>0</sub>(''N''))}} is spanned by ([[Weakly holomorphic modular form|weakly holomorphic]]) eta quotients,<ref name="RW"/> and {{math|''M''<sub>''k''{{prime}}−''k''</sub>(Γ<sub>0</sub>(''N''))}} contains an eta quotient {{mvar|η<sub>g</sub>}}.

|Take a modular form {{mvar|f}} with weight {{mvar|k}} that is not in the span of our computed eta quotients, and compute {{math|''f'' ''η<sub>g</sub>''}} as a linear combination of eta-quotients in {{math|''M''<sub>''k''{{prime}}</sub>(Γ<sub>0</sub>(''N''))}} and then divide out by {{mvar|η<sub>g</sub>}}. The result will be an expression of {{mvar|f}} as a linear combination of eta quotients as desired. Repeat this until a basis is formed.
}}
A collection of over 6300 product identities for the Dedekind eta function in a canonical, standardized form is available at the Wayback machine<ref>{{cite web | url=http://eta.math.georgetown.edu/index.html | archive-url=https://web.archive.org/web/20190709153048/http://eta.math.georgetown.edu/index.html | archive-date=2019-07-09 | title=Dedekind Eta Function Product Identities by Michael Somos }}</ref> of Michael Somos' website.