Lattice theta functional equation (${\Theta }_{\Lambda }(\tau )=(-i\tau {)}^{-n/2}m(\Lambda {)}^{-1}{\Theta }_{{\Lambda }^{\vee }}(-1/\tau )$)

For a full-rank Euclidean lattice, ${\Theta }_{\Lambda }(\tau )={\sum }_{x\in \Lambda }{e}^{\pi i|x{|}^2\tau }$ obeys ${\Theta }_{\Lambda }(\tau )=(-i\tau {)}^{-n/2}m(\Lambda {)}^{-1}{\Theta }_{{\Lambda }^{\vee }}(-1/\tau )$. Apply the Poisson summation formula for a Euclidean lattice and the complex Gaussian Fourier transform. The identity holds without assuming an integral or self-dual lattice.

For a full Euclidean lattice and a positive definite real symmetric matrix $A$, the Gaussian theta sum obeysUse the Fourier transform kernel $e^{-2\pi ix\cdot z}$, the identity $\widehat{e^{-\pi x^{T}Ax}}(z)=(\det A{)}^{-1/2}{e}^{-\pi z^T{A}^{-1}z}$, and the Poisson summation formula for a Euclidean lattice. No integrality or self-duality of the Euclidean lattice is assumed.