Abstract
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.
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