Critical modular lattices in the Gaussian core model

We discuss the local analysis of Gaussian potential energy of modular lattices. We present examples of $2$-modular lattices – such as the $16$-dimensional Barnes-Wall lattice – and $3$-modular lattices – such as the $12$-dimensional Coxeter-Todd lattice – that are locally universally optimal among lattices (in the sense of Cohn and Kumar). We also provide other $2$- and $3$-modular lattices that are not locally universally optimal, or not even critical in the Gaussian core model.