Critical modular lattices in the Gaussian core model

Abstract

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.

32 pages, (v2): Computational results reported in Section 5 were wrong in (v1), due to an indexing mistake. Now corrected and thoroughly checked, relevant parts completely rewritten