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A note on unimodular lattices with trivial automorphism groups. (English) Zbl 1364.11131

In [Mem. Am. Math. Soc. 429, 70 p. (1990; Zbl 0702.11037)], E. Bannai proved, using Siegel’s mass formula, that there exist an odd unimodular lattice and an even unimodular lattice with trivial automorphism groups provided the dimension \(n \geq 43\) or \(n = 8k \geq 144\), respectively. From the values of the masses of positive definite unimodular lattices given in the book by J. H. Conway and N. J. A. Sloane [Sphere packings, lattices and groups. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. 3rd ed. New York, NY: Springer (1999; Zbl 0915.52003)], or by O. D. King in [Math. Comput. 72, No. 242, 839–863 (2003; Zbl 1099.11035)], there exist no odd unimodular lattice with \(n < 28\) and even unimodular lattice with \(n <32\) having trivial automorphism groups. In [in: Résaux euclidiens, designs sphériques et formes modulaires. Autour des travaux de Boris Venkov. Genève: L’Enseignement Mathématique. 212–267 (2001; Zbl 1139.11319)] R. Bacher and B. Venkov proved that there is no unimodular lattice with a trivial automorphism group for \(n= 28\). Y. Mimura in [in: Algebra and topology, Proc. 5th Math. Workshop, Taejon/Korea 1990, Proc. KIT Math. Workshop 5, 91–95 (1990; Zbl 0735.11026)] constructed explicit examples of odd unimodular lattices of dimension \(36, 40\) and an even unimodular lattice of dimension \(64\) with trivial automorphism groups by using the neighbor method of Kneser. Using the same method with the aid of computer, R. Bacher in [Int. Math. Res. Not. 1994, No. 2, 91–95 (1994; Zbl 0817.11034)] obtained an odd lattice with \(n = 29\) and an even lattice with \(n= 32\) having trivial automorphism groups. M. Harada [Discrete Math. 239, No. 1–3, 121–125 (2001; Zbl 0988.94026)] also gave a \(32\)-dimensional odd unimodular lattice with a trivial automorphism group. In this paper, a different effective construction method, has been presented to obtain odd positive definite unimodular lattices of dimensions \(29 \leq n \leq 42\) and even positive definite unimodular lattices of dimensions \(n = 32, 40\) with trivial automorphism groups. Combining these results with Bannai’s and Bacher and Venkov’s results it is obtained that there exists a unimodular lattice with a trivial automorphism group if and only if the dimension \(n > 28\).

MSC:

11H56 Automorphism groups of lattices
11H06 Lattices and convex bodies (number-theoretic aspects)

Software:

Magma
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References:

[1] Bacher, R., Unimodular lattices without nontrivial automorphisms, Int. Math. Res. Not. IMRN, 19, 888-896, (1994) · Zbl 0817.11034
[2] Bacher, R.; Venkov, B., Rseaux entiers unimodulaires sans racine en dimension 27 et 28, (Rseaux euclidiens, designs sphriques et formes modulaires, Monogr. Enseign. Math., vol. 37, (2001), L’Enseignement Mathmatique Geneva), 212-267 · Zbl 1139.11319
[3] Bannai, E., Positive definite unimodular lattices with trivial automorphism groups, Mem. Amer. Math. Soc., 85, 91-95, (1990)
[4] Bosma, W.; Cannon, J.; Ployoust, C., The magma algebra system,1. the user langage, J. Symbolic Comput., 24, 235-365, (1997)
[5] Conway, J. H.; Sloane, N. J., Sphere Packings, Lattices and Groups, (1999), Springer-Verlag New York · Zbl 0915.52003
[6] Gerstein, L., Nearly unimodular quadratic forms, Ann. of Math., 142, 597-610, (1995) · Zbl 0842.11012
[7] Harada, M., An extremal ternary self-dual [28,14,9] code with a trivial automorphism group, Discrete Math., 239, 121-1255, (2001) · Zbl 0988.94026
[8] King, O., A mass formula for unimodular lattices with no roots, Math. Comp., 72, 839-863, (2003) · Zbl 1099.11035
[9] Mimura, Y., Explicit examples of unimodular lattices with the trivial automorphism group, (Algebra and Topology, vol. 19, (1990), Korea Adv. Inst. Sci. Tech. Taejon), 91-95 · Zbl 0735.11026
[10] Newman, M., Tridiagonal matrixes, Linear Algebra Appl., 201, 51-55, (1994)
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