A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916071993245696 |
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| author | Kominers, Scott Duke |
| author_facet | Kominers, Scott Duke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01347 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$ Kominers, Scott Duke Number Theory Functional Analysis Metric Geometry Primary 11H06, 11F27, Secondary 52C17, 11F11 G.2; E.4 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. |
| title | A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$ |
| topic | Number Theory Functional Analysis Metric Geometry Primary 11H06, 11F27, Secondary 52C17, 11F11 G.2; E.4 |
| url | https://arxiv.org/abs/2606.01347 |