Geodesic
Rings of integers in number fields and root lattices
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 58-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{DANMA_2020_492_a11,
     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},
     year = {2020},
     volume = {492},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_492_a11/}
}
TY  - JOUR
AU  - V. L. Popov
AU  - Yu. G. Zarhin
TI  - Rings of integers in number fields and root lattices
JO  - Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
PY  - 2020
SP  - 58
EP  - 61
VL  - 492
UR  - http://geodesic.mathdoc.fr/item/DANMA_2020_492_a11/
LA  - ru
ID  - DANMA_2020_492_a11
ER  - 
%0 Journal Article
%A V. L. Popov
%A Yu. G. Zarhin
%T Rings of integers in number fields and root lattices
%J Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
%D 2020
%P 58-61
%V 492
%U http://geodesic.mathdoc.fr/item/DANMA_2020_492_a11/
%G ru
%F DANMA_2020_492_a11
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/

[1] Andrade A.A., Interlando J.C., “Rotated ${{\mathbb{Z}}^{n}}$-lattices via real subfields of $\mathbb{Q}({{\zeta }_{{{{2}^{r}}}}})$”, TEMA - Tend. em Mat. Apl. e Comput., 20:3 (2019), 445–456 | DOI | MR

[2] Bayer-Fluckiger E., “Lattices and number fields”, Contemporary Math., 241, 1999, 69–84 | DOI | MR | Zbl

[3] Bayer-Fluckiger E., “Upper bounds for Euclidean minima of algebraic number fields”, J. Number Theory, 121 (2006), 305–323 | DOI | MR | Zbl

[4] Bayer-Fluckiger E., Maciak P., “Upper bounds for the Euclidean minima of abelian fields”, J. Théorie des Nombres de Bordeaux, 27 (2015), 689–697 | DOI | MR | Zbl

[5] Conway J.H., Sloane N.J.A., Sphere Packing, Lattices and Groups, Springer-Verlag, N.Y., 1988 | MR

[6] Martinet J., Perfect Lattices in Euclidean Spaces, Springer-Verlag, B., 2003 | MR | Zbl

[7] Milnor J., Husemoller D., Symmetric Bilinear Forms, Springer-Verlag, B., 1973 | MR | Zbl

[8] Witt E., “Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe”, Abh. Math. Sem. Univ. Hamb., 14 (1941), 289–322 | DOI | MR | Zbl