Evaluation of mock modular forms at elliptic points

Question

The holomorphic function $$F(\tau)=-\frac{1}{\vartheta_4(\tau)}\sum_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$ is a mock modular form for the congruence subgroup $\Gamma^0(4)$ of $\text{SL}(2,\mathbb Z)$ with shadow $\eta^3$. Here, $q=e^{2\pi i\tau}$, $\vartheta_4(\tau)=\sum_{n\in\mathbb Z}(-1)^n q^{n^2/2}$ is a Jacobi theta function and $\eta$ is the Dedekind eta function. The $q$-series is the OEIS sequence A256209.

As a consequence, the sum of $F$ and a non-holomorphic period integral $$ \hat F(\tau,\bar\tau)=F(\tau)-\frac i2\int_{-\bar\tau}^{i\infty}\frac{\eta(w)^3}{\sqrt{-i(w+\tau)}}\mathrm dw. $$ transforms as a non-holomorphic modular form of weight $(\tfrac12,0)$ for $\Gamma^0(4)$, and it is clear that $\partial_{\bar\tau}\hat F(\tau,\bar\tau)=-\frac{i}{2\sqrt{2y}}\overline{\eta(\tau)^3}$.

I am interested in special values of $F$ and $\hat F$, such as at the elliptic points of $\text{SL}(2,\mathbb Z)$. For instance, numerically I find that $$\hat F(i,\bar i)=\frac{\vartheta_4(i)}{2^{\frac54}}, \\ \hat F(\alpha,\bar\alpha)=\frac{e^{-\frac{\pi i}{4}}\vartheta_4(\alpha)}{2\sqrt3},$$ where $\alpha=e^{2\pi i/3}$, while $\hat F(i+1,\overline{i+1})=0$. The latter can be proven by realising that $\hat F$ is more precisely a non-holomorphic modular form for $\Gamma^0(2)$, whose multiplier system evaluated at the elliptic fixed point $1+i$ of $\Gamma^0(2)$ is $-1$.

My questions:

• Are explicit values of mock modular forms and their completions at elliptic points known in the literature?
• Is there a general strategy to calculate series and period integrals, such as those involved in $\hat F(i,\bar i)$ and $\hat F(\alpha,\bar\alpha)$?

Answer 1

The holomorphic function $F$ is related to the $q$-series $H^{(2)}$ of Mathieu moonshine in a simple way. We have \begin{equation} \begin{aligned} H^{(2)}(\tau)&=2\frac{\vartheta_2(\tau)^4-\vartheta_4(\tau)^4}{\eta(\tau)^3}-\frac{24}{\vartheta_3(\tau)}\sum_{n\in\mathbb Z}\frac{q^{\frac{n^2}{2}-\frac18}}{1+q^{n-\frac12}}\\ &=2q^{-\frac18}\left(-1+45q+231q^2+770q^3+2277q^4+\dots\right) \end{aligned} \end{equation} The $q$-series is the OEIS sequence A169717, and the Fourier coefficients are sums of dimensions of irreducible representations of the sporadic group $M_{24}$. It relates to $F$ as \begin{equation}\label{FH_relation} F(\tau)=\frac{1}{24}\left( H^{(2)}(\tau)+2\frac{\vartheta_2(\tau)^4+\vartheta_3(\tau)^4}{\eta(\tau)^3}\right). \end{equation} Since both $H^{(2)}$ and $F$ are mock modular forms with shadow $\eta^3$, their non-holomorphic completions $\widehat H^{(2)}$ and $\widehat F$ are simply related by \begin{equation}\label{FhatHhat} \widehat F(\tau)=\frac{1}{24}\left( \widehat H^{(2)}(\tau)+2\frac{\vartheta_2(\tau)^4+\vartheta_3(\tau)^4}{\eta(\tau)^3}\right). \end{equation} Unlike $ \widehat F$, $\widehat H^{(2)}$ is a non-holomorphic modular function for $\text{SL}(2,\mathbb Z)$, and transforms under the generators as \begin{equation} \begin{aligned} \widehat H^{(2)}(\tau+1,\bar\tau+1)&=e^{-\frac{\pi i}{4}}\widehat H^{(2)}(\tau,\bar\tau), \\ \widehat H^{(2)}(-1/\tau,-1/\bar\tau)&=-\sqrt{-i\tau}\widehat H^{(2)}(\tau,\bar\tau). \end{aligned} \end{equation} From the phases of the transformations under the elliptic elements of $\text{SL}(2,\mathbb Z)$ stabilising the elliptic fixed points $i$ and $\alpha$, we easily find \begin{equation}\begin{aligned} \widehat H^{(2)}(i,\bar i)&=0, \\ \widehat H^{(2)}(\alpha,\bar \alpha)&=0. \end{aligned}\end{equation} Thus at $\tau=i$ and $\tau=\alpha$ the completion $ \widehat H^{(2)}(\tau,\bar\tau)$ evaluates to the modular "difference" between $ \widehat H^{(2)}$ and $\widehat F$, which can be determined using the Chowla-Selberg formula.