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.

💡 Research Summary

The paper investigates the local behavior of the Gaussian core model energy for modular lattices, extending previous work that focused mainly on even unimodular lattices such as E₈ and the Leech lattice. The Gaussian core model assigns to a point configuration the energy
(E(\alpha,L)=\sum_{x\in L\setminus{0}}e^{-\alpha|x|^{2}})
for a lattice (L) of unit point density and a parameter (\alpha>0). A configuration is called universally optimal if it minimizes this energy for every completely monotonic potential, which is equivalent to minimizing the Gaussian energy for all (\alpha).

The authors turn to (\ell)-modular lattices, defined by the existence of a similarity (\sigma) of norm (\ell) such that (L=\sigma(L^{#})) (equivalently (L\cong\sqrt{\ell},L^{#})). For such lattices the theta series (\Theta_{L}(\tau)=\sum_{v\in L}e^{\pi i\tau|v|^{2}}) is a modular form of weight (k=n/2) for the Fricke group (\Gamma^{}(\ell)) with a character (\chi). When (1+\ell) divides 24 (the primes (\ell=2,3,5,7,11,23)), the graded algebra of these modular forms is generated by a single theta series (\Theta) and a cusp form (\Delta_{\ell}), i.e. (\mathcal{M}(\Gamma^{}(\ell),\chi)=\mathbb{C}