Spherical designs and heights of Euclidean lattices (Q402650)

Let \(\Lambda\) be a full-rank lattice in \(\mathbb{R}^n\) with determinant 1. The height function \(\Lambda\to h(\Lambda)\) can be defined as NEWLINE\[NEWLINEZ'(\Lambda^*,0)+2\log(2\pi),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEZ(\Lambda,s)=\sum_{0\neq x\in\Lambda} \frac{1}{||x||^{2s}}NEWLINE\]NEWLINE is an Epstein zeta function, and NEWLINE\[NEWLINE\Lambda^*=\{y\in\mathbb{R}^n:(x\cdot y)\in\mathbb{Z} \text{ for all } x\in\Lambda\}NEWLINE\]NEWLINE is the dual for \(\Lambda\). The height function can also be defined in terms of eigenvalues of the Laplace operator on \(\mathbb{R}^n/\Lambda\).NEWLINENEWLINEA lattice \(\Lambda\) is called fully critical lattice if it is a stationary point of the height function. The main result of the paper states that a lattice is fully critical if and only if all its layers hold a spherical 2-design.NEWLINENEWLINEUsing this criterion and modular forms techniques, the authors find all fully critical lattices in dimensions \(2\leq n\leq 6\). It is also proved that the root lattices \(A_n\), \(D_n\), \(E_6\), \(E_7\), and \(E_8\) are fully critical.