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## 第95回 2019年9月6日 14:00〜15:30

Qianqian Yang（中国科学技術大学，東北大学）「The $2$-integrability of integral lattices of rank $12$」

Given a positive integer $s$, an integral lattice $\Lambda$ of rank $n$ is $s$-integrable if there exist $\mathbf{v}_1,\ldots,\mathbf{v}_{\ell}\in \mathbb{Z}^n$ such that the Gram matrix of $\Lambda$ with respect to some basis is the matrix $\sum_{i=1}^{\ell}\mathbf{v}_i\mathbf{v}_i^T$. Let
\[\phi(s)=\min\{r\mid \text{ there exists an integral lattice of rank $r$ which is not $s$-integrable}\}.\]
In 1988, Conway and Sloane showed that $\phi(2)=12$ and every unimodular lattice of rank $14$ is $2$-integrable. So if an integral lattice of rank $12$ can be embedded in a unimodular lattice of rank $14$, then it must be $2$-integrable. In this talk, I will show our result and give the sufficient and necessary condition to embed an integral lattice of rank $n$ in a unimodular lattice of rank $n+2$. As a corollary of our result, I will show that if the determinant of an integral lattice of rank $12$ satisfies certain condition, it is $2$-integrable.