Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 3 / c / Solution

The Poisson summation formula for the Gaussian gives the transformations of the Jacobi theta functionwhere the square root is the holomorphic branch on $\mathfrak{h}$. Raising to the $k$th power gives invariance under $T^2$. If $k=8m$, thenso ${\theta }^k(-1/\tau )={\tau }^{k/2}{\theta }^k(\tau )$. Since $T^2$ and $S$ generate the theta group, this is the weight-$k/2$ transformation law on $\Gamma$.

The defining series converges locally uniformly, so ${\theta }^k$ is holomorphic on $\mathfrak{h}$, and its expansion at infinity has no negative powers of $e^{\pi i\tau }$. Applying Poisson summation to the shifted Gaussian $e^{-\pi t(n+1/2{)}^2}$ gives the corresponding nonnegative-power expansion at the remaining cusp. Hence ${\theta }^k$ is holomorphic at every cusp and is a modular form of weight $k/2$ and level $\Gamma$.