A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$

Abstract

The integer lattice $\mathbb{Z}^n$ is conjectured to maximize the Gaussian mass $Θ_L(t)=\sum_{x\in L}e^{-t\|x\|^2}$ over the set of stable lattices in $\mathbb{R}^n$, for every $t>0$. We prove this sharp inequality for every integral unimodular lattice $L$ of rank $n\leq 32$, with equality only at $L\cong\mathbb{Z}^n$, and furthermore obtain the strict inequality for every even unimodular lattice of rank $40$. The proof does not use the classification of unimodular lattices in these ranks; rather, it parametrizes integral unimodular theta series as polynomials in the modular function $u=Δ_8/\vartheta_3^8\in(0,1/64]$, with the few coefficients that arise controlled by norm-$1$ splitting, ADE root counts, and shadow positivity.