We study the zeros of theta functions $\Theta_{ \Gamma_{ 4k}}$ associated with the lattices $\Gamma_{ 4k}$, 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 $\Gamma_{ 4k}$ is an even lattice, we show that the zeros all lie on the line $\Re z =\frac{ 1}{ 2}$ in the fundamental domain and prove that the zeros are equidistributed with respect to an explicit probability measure on the line $\Re z = \frac{ 1}{ 2}$. However, when the $\Gamma_{ 4k}$ is an odd lattice, there are no zeros on the line $\Re z =\frac{ 1}{ 2}$, only exponentially close to it. Our argument relies on representing $\Theta_{ \Gamma_{ 4k}}$ 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.
Categories: math.NT math.CA