Estimating two dimensional theta function

Question

My feeling is that this should be written somewhere but I don't know what to search for.

Let $Q(x,y)$ be a binary quadratic form over $\mathbb{C}$, with $\operatorname{Re}(Q)$ positive definite. Then we can form the series $$ \sum_{u,v\in\mathbb{Z}} e^{-Q(u,v)} $$

I need to estimate this sum, especially in the case where $\operatorname{Im}(Q)$ is a fixed positive definite quadratic form and $\operatorname{Re}(Q)$ is close to a degenerate quadratic form of rank $1$. That is, $Q$ is of the form $$ Q(x,y) = i\left(ax^2+bxy+cy^2\right) + \left(\alpha x + \beta y\right)^2 + \epsilon \left(-\beta x+\alpha y\right)^2 $$ where $a,b,c,\alpha,\beta,\epsilon\in\mathbb{R}$, $ac-b^2>0$, $(\alpha,\beta)\neq(0,0)$, $\epsilon>0$ and let $\epsilon\to0$

My feeling is that, in the above case, if we fix $a,b,c,\alpha,\beta$ and let $\epsilon\to0$ then the sum should be something like $O\left(\epsilon^{-\frac12}\right)$ but I need to know how the constant in the big-O depends on $a,b,c,\alpha,\beta$.

Answer 1

Using the Poisson summation formula, with $P=\pmatrix{\alpha&\beta\\ -\beta&\alpha}$ and $D=\pmatrix{1/\pi^{1/2}&0\\0&\epsilon^{1/2}/\pi^{1/2}}$ then $$|\sum_{n\in \Bbb{Z}^2} e^{-Q(n)}|\le \sum_{n\in \Bbb{Z}^2} e^{-\pi \|DPn\|^2}=|\det(DP)|^{-1} \sum_{n\in \Bbb{Z}^2} e^{-\pi \| D^{-1} P^{-\top} n\|^2}$$ $$\le |\det(DP)|^{-1} \sum_{n\in \Bbb{Z}^2} e^{-\pi \| P^{-\top} n\|^2}$$

Which is a $O(\epsilon^{-1/2})$ bound depending only on $\alpha,\beta$.

Not sure what is the impact of the imaginary part.