On Skew-holomorphic Jacobi Forms

Abstract

Let k, m be positive integers and let k be odd. Let µ (mod 2m) be an integer. The theta function

$${\theta _{m,\mu }}(\tau ,z): = \sum\limits_{\begin{array}{*{20}{c}} {r \in \mathbb{Z}} \\ {r \equiv \mu (\bmod 2m)} \end{array}} {e\left( {\frac{{{r^2}}}{{4m}}\tau + rz} \right)} ,$$

(1)

where e(w) := e ^2πiw, w ∈ ℂ, satisfies the heat equation

$$\left( {8\pi im\frac{\partial}{{\partial {\tau ^2}}}{\text{ - }}\frac{{{\partial ^2}}}{{\partial {z^2}}}} \right){\theta _{m,\mu }}\left( {r,z} \right) = 0,$$

((2))

and further it satisfies the following transformation law:

$${\theta _{m,\mu }}\left( {\tau ,z + s\tau + t} \right) = {e^{ - m}}\left( {{s^2}\tau + 2sz} \right){\theta _{m,\mu }}\left( {\tau ,z} \right),$$

((3))

where e ^m(w) := e ^2πimw, w ∈ ℂ. The Poisson summation formula gives

$${\theta _{m,\mu }}\left( { - \frac{1}{\tau },\frac{z}{\tau }} \right) = \sqrt {\frac{\tau }{{2mi}}} {e^m}\left( {{z^2}/\tau } \right)\sum\limits_{\lambda \,\bmod \,2m} {{e_{2m}}} \left( { - \lambda \mu } \right){\theta _{m,\lambda }}\left( {\tau ,z} \right)$$

((4))