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Modular form

Lattice model of a modular form

{title2=$\theta_k(\Lambda)$}

For an even nonnegative integer $k$,
$$
\theta_k(\Lambda)=\sum_{\lambda\in\Lambda}\lambda^ke^{-\pi|\lambda|^2}.
$$
Poisson summation and the Fourier eigenfunction identity for $(x+iy)^ke^{-\pi(x^2+y^2)}$ give
$$
\theta_k(\Lambda)=(-i)^k m(\Lambda)^{-1}\theta_k(\Lambda^\vee).
$$


Weighted Gaussian theta sum of a complex lattice

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 137 / 4 / b / Solution

Weighted Gaussian theta sum of a complex lattice ( $θ_{k} (Λ)$ )

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Weighted Gaussian theta sum of a complex lattice (θk​(Λ))

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Weighted Gaussian theta sum of a complex lattice ( $θ_{k} (Λ)$ )

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