Geodesic
Rings of integers in number fields and root lattices
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 58-61.

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This paper investigates whether a root lattice can be similar to the lattice $\mathscr{O}$ of all integer elements of a number field $K$ endowed with the inner product $(x,y):=\operatorname{Trace}_{K/\mathbb{Q}}(x\cdot\theta(y))$, where $\theta$ is an involution of the field $K$. For each of the following three properties (1), (2), (3), a classification of all the pairs $K$, $\theta$ with this property is obtained: (1) $\mathscr{O}$ is a root lattice; (2) $\mathscr{O}$ is similar to an even root lattice; (3) $\mathscr{O}$ is similar to the lattice $\mathbb{Z}^{[K:\mathbb{Q}]}$. The necessary conditions for similarity of $\mathscr{O}$ to a root lattice of other types are also obtained. It is proved that $\mathscr{O}$ cannot be similar to a positive definite even unimodular lattice of rank $\le48$, in particular, to the Leech lattice.
Keywords: number field, ring of integers, root lattice. 
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     author = {V. L. Popov and Yu. G. Zarhin},
     title = {Rings of integers in number fields and root lattices},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {58--61},
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     volume = {492},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_492_a11/}
}
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V. L. Popov; Yu. G. Zarhin. Rings of integers in number fields and root lattices. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 58-61. http://geodesic.mathdoc.fr/item/DANMA_2020_492_a11/

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