Spherical designs and heights of Euclidean lattices

We study the connection between the theory of spherical designs and the question of extrema of the height function of lattices. More precisely, we show that a full-rank n-dimensional Euclidean lattice Λ, all layers of which hold a spherical 2-design, realises a stationary point for the height h(Λ), which is defined as the first derivative at the point 0 of the spectral zeta function of the associated flat torus ζ(Rn/Λ). Moreover, in order to find out the lattices for which this 2-design property holds, a strategy is described which makes use of theta functions with spherical coefficients, viewed as elements of some space of modular forms. Explicit computations in dimension n⩽7, performed with Pari/GP and Magma, are reported.