Embedding theory of lattices and its application for 2-integrable lattices

2021/04/09 by Qianqian Yang, Yang, Qianqian, Kiyoto Yoshino +1
Mathematics · #11E08 #11E25 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11E08 #msc:11E25

paper · pdf · doi:10.48550/arxiv.2104.04177

16 pages

arxiv created 2021/04/09 · arxiv updated 2021/04/12

Abstract

For a positive integer s, a lattice L is said to be s-integrable if √(s)⋅ L is isometric to a sublattice of ℤⁿ for some integer n. Conway and Sloane found two minimal non 2-integrable lattices of rank 12 and determinant 7 in 1989. We find two more ones of rank 12 and determinant 15. Then we introduce a method of embedding a given lattice into a unimodular lattice, which plays a key role in proving minimality of non 2-integrable lattices and finding candidates for non 2-integrable lattices.

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