Dedekind Eta Database

Introduction

The Dedekind eta function, \( \eta(\tau) \), is a modular form with the following properties:

• Holomorphic on the upper half-plane: \( \text{Im}(\tau) > 0 \)
• q-expansion: \[ \eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 - q^n), \quad q = e^{2 \pi i \tau} \tag{1} \]

  From (1), it is evident that,

  \[ \eta(iy) \in \mathbb{R}^{+}, \quad e^{\frac{-i\pi}{24}}\eta\left(\frac{1+iy}{2}\right) \in \mathbb{R}^{+}, \tag{2} \] for \(y>0\).

• Modular transformations:

\[ \eta(\tau + 1) = e^{\pi i /12} \eta(\tau) \tag{3} \]

\[ \eta(-1/\tau) = \sqrt{-i\tau} \, \eta(\tau) \tag{4} \]

If we choose \( u(\tau) = \left( \frac{\eta(2\tau)}{\eta(\tau)} \right)^{24} \) to be a Hauptmodul for \( \Gamma_0(2) \), then it is related to the j-invariant as:

\[ j(\tau) = \frac{(1 + 256 \cdot u(\tau))^3}{u(\tau)} \tag{5} \]

• Work of Muzaffar and Williams ^{[1], [2]}

Let \(\lambda \in \mathbb{N}, d=-4\lambda, \) such that \(d\equiv 0 \text{ or } 1\pmod{4}\). Choose largest \(f\) such that \(f^2|d, d/f^2\equiv 0 \text{ or } 1\pmod{4}\). Set \(\Delta=d/f^2\). Let \(v_p(f)\) be a non-negative integer such that \(p^{v_p(f)}|p\) but \(p^{v_p(f)+1}\nmid p\). Then, define

\[ \alpha_p(\Delta, f)=\frac{(p^{v_p(f)}-1)\left(1-\left(\frac{\Delta}{p}\right) \right)}{ p^{v_p(f)-1}(p-1) \left(p - \left(\frac{\Delta}{p}\right)\right)}, \]

where \( \left(\frac{\Delta}{p}\right)\) is the Legengre symbol. Let \(w(d)\) denote the number of autumorphs of primitive, positive-definite, binary quadratic form of discriminant \(d\). Then, \(w(d)=\begin{cases} 6, &\text{ if } d=-3\\ 4, &\text{ if } d=-4\\ 2, &\text{ if } d<-4 \end{cases}\). The positive-definite, primite, integral, binary quadratic form \(ax^2+bxy+cy^2\), denoted by \((a,b,c)\),with discriminant \(d=b^2-4ac\), and class

\[ [a,b,c]= \{ a(p,r), b(p,q,r,s), c(q,s) \mid p,q,r,s \in \mathbb{Z}, ps-qr=1 \}, \] where \(a(p,r)=ap^2+bpr+cr^2, b(p,q,r,s)=2apq+bps+bqr+2crs, c(q,s)=aq^2+bqs+cs^2\), forms a finite abelian group under Gaussian composition, denoted by \(H(d)\) and the order of \(H(d)\) is denoted by \(h(d)\). Moreover, there exists positive integers \(h_1,\cdots, h_v\) and generators \(A_1, \cdots A_v\in H(d)\) such that \(h_1h_2\cdots h_v=h(d)\),\(1 < h_1 \mid h_2 \cdots \mid h_v\) , and \(\text{ord}(A_i)=h_i\). For each \(K\in H(d))\), there exists unique integers \(k_1\cdots k_v\) with \(0\leq k_j\leq h_j\) such that \(K=A_1^{k_1}\cdots A_v^{k_v}\). Set, \(\text{ind}_{A_j}(K):=k_j\). Then we define \(\chi\) on \(H(d) \times H(d)\) as

\[ \chi(K,L)=e^{2\pi i \sum_{j=1}^{v}\frac{\text{ind}_{A_j}(K) \text{ind}_{A_j}(L)}{h_j}}. \] For \(K=[a,b,c]\in H(d), K^{-1}=[a,-b,c]\) and define \(K_p=\left[p, h_1, \frac{h_1^2-d}{4p}\right]\), where \(h_i^2\equiv d\pmod(4p), h_1 < h_2\). Define \(H_L(n)=\text{ card } \{h \mid 0\leq h <2n, h^2\equiv d\pmod{4n}, \left[n,h,\frac{h^2-d}{4n}\right]=L\}\). Then, we define,

\[ \eta(\sqrt{-\lambda})=2^{-3/4}\pi^{-1/4}\lambda^{-1/4}\prod_{p|f}p^{\alpha_p(\Delta,f)/4}\left( \prod_{m=1}^{|\Delta|}\Gamma\left(\frac{m}{|\Delta|}\right)^\left(\frac{\Delta}{m}\right) \right)^{\frac{w(\Delta)}{8h(\Delta)}}e^{-E(K,d)} \tag{6} \]