Zeros of theta functions associated with self-dual lattices

We study the zeros of theta functions ${\mathrm{\Theta }}_{{\mathrm{\Gamma }}_{4k}}$ associated with the lattices ${\mathrm{\Gamma }}_{4k}$, a family of self-dual lattices generalizing the ${\mathsf{E}}_8$ lattice. Our results show two different behaviors of the zeros according to the lattice parity: When ${\mathrm{\Gamma }}_{4k}$ is an even lattice, we show that the zeros all lie on the line $\mathrm{\Re }z=\frac{1}{2}$ in the fundamental domain and prove that the zeros are equidistributed with respect to an explicit probability measure on the line $\mathrm{\Re }z=\frac{1}{2}$. However, when the ${\mathrm{\Gamma }}_{4k}$ is an odd lattice, there are no zeros on the line $\mathrm{\Re }z=\frac{1}{2}$, only exponentially close to it. Our argument relies on representing ${\mathrm{\Theta }}_{{\mathrm{\Gamma }}_{4k}}$ as a polynomial in the modular $\lambda$-function. We then study the zeros of this polynomial and exploit some conformal properties of $\lambda$ to get our results.